How Do You Know When to Use the Limit Comparison Test

Learning Objectives

• 5.four.1 Use the comparing test to test a series for convergence.
• 5.4.2 Use the limit comparison test to determine convergence of a series.

We have seen that the integral test allows us to decide the convergence or divergence of a serial by comparison information technology to a related improper integral. In this department, we show how to employ comparing tests to make up one's mind the convergence or divergence of a series by comparing it to a series whose convergence or difference is known. Typically these tests are used to make up one's mind convergence of series that are like to geometric series or p-series.

Comparison Exam

In the preceding two sections, we discussed two large classes of serial: geometric series and p-series. Nosotros know exactly when these series converge and when they diverge. Hither nosotros prove how to use the convergence or divergence of these series to testify convergence or divergence for other serial, using a method called the comparison test.

For example, consider the series

$${\displaystyle \sum }_{n=\mathrm{ane}}^{\infty }\frac{1}{{\mathrm{due\; north}}^2+1}.$$

This series looks similar to the convergent serial

$${\displaystyle \sum }_{\mathrm{north}=\mathrm{one}}^{\infty }\frac{\mathrm{i}}{n^2}.$$

Since the terms in each of the serial are positive, the sequence of partial sums for each serial is monotone increasing. Furthermore, since

$$0<\frac{1}{n^2+1}<\frac{\mathrm{ane}}{n^2}$$

for all positive integers $\mathrm{due\; north},$ the $k\text{th}$ partial sum $S_k$ of ${\displaystyle \sum }_{\mathrm{northward}=1}^{\infty }\frac{1}{n^2+\mathrm{ane}}$ satisfies

$$S_g={\displaystyle \sum }_{n=\mathrm{one}}^k\frac{1}{n^{\mathrm{ii}}+\mathrm{i}}<{\displaystyle \sum }_{\mathrm{north}=1}^{\mathrm{thousand}}\frac{\mathrm{one}}{n^2}<{\displaystyle \sum }_{n=1}^{\infty }\frac{1}{n^2}.$$

(See Effigy five.16(a) and Table v.ane.) Since the series on the correct converges, the sequence $\left\{{\mathrm{South}}_g\right\}$ is bounded to a higher place. We conclude that $\left\{{\mathrm{Due\; south}}_{\mathrm{1000}}\right\}$ is a monotone increasing sequence that is bounded to a higher place. Therefore, by the Monotone Convergence Theorem, $\left\{S_k\right\}$ converges, and thus

$${\displaystyle \sum }_{\mathrm{due\; north}=1}^{\infty }\frac{1}{n^2+\mathrm{one}}$$

converges.

Similarly, consider the series

$${\displaystyle \sum }_{n=1}^{\infty }\frac{1}{n-1\text{/}\mathrm{two}}.$$

This series looks similar to the divergent series

$${\displaystyle \sum }_{n=1}^{\infty }\frac{\mathrm{ane}}{n}.$$

The sequence of partial sums for each series is monotone increasing and

$$\frac{1}{n-1\text{/}2}>\frac{\mathrm{one}}{n}>0$$

for every positive integer $\mathrm{northward}.$ Therefore, the $k\text{th}$ partial sum ${\mathrm{Southward}}_{\mathrm{grand}}$ of ${\displaystyle \sum }_{\mathrm{north}=1}^{\infty }\frac{1}{n-\mathrm{ane}\text{/}2}$ satisfies

$${\mathrm{South}}_{\mathrm{thousand}}={\displaystyle \sum }_{\mathrm{north}=1}^{\mathrm{thou}}\frac{1}{n-1\text{/}\mathrm{ii}}>{\displaystyle \sum }_{n=\mathrm{one}}^k\frac{1}{n}.$$

(See Figure 5.xvi(b) and Tabular array 5.2.) Since the serial ${\displaystyle \sum }_{\mathrm{north}=1}^{\infty }1\text{/}n$ diverges to infinity, the sequence of partial sums ${\displaystyle \sum }_{\mathrm{due\; north}=1}^{k}1\text{/}n$ is unbounded. Consequently, $\left\{S_k\right\}$ is an unbounded sequence, and therefore diverges. We conclude that

$${\displaystyle \sum }_{n=\mathrm{ane}}^{\infty }\frac{1}{n-\mathrm{i}\text{/}2}$$

diverges.

Figure v.sixteen (a) Each of the partial sums for the given series is less than the corresponding fractional sum for the converging $p-\text{series}.$ (b) Each of the partial sums for the given serial is greater than the corresponding partial sum for the diverging harmonic serial.

$k$	$1$	$2$	$\mathrm{three}$	$4$	$\mathrm{five}$	$\mathrm{vi}$	$7$	$\mathrm{viii}$
${\displaystyle \sum _{n=1}^k\frac{1}{{\mathrm{northward}}^2+\mathrm{ane}}}$	$0.5$	$0.7$	$\mathrm{0.viii}$	$0.8588$	$0.8973$	$0.9243$	$0.9443$	$0.9597$
${\displaystyle \sum _{\mathrm{northward}=1}^k\frac{\mathrm{one}}{n^2}}$	$1$	$\begin{array}{l}\mathrm{ane.25}\hfill \end{array}$	$\mathrm{ane.3611}$	$1.4236$	$1.4636$	$\mathrm{one.4914}$	$1.5118$	$1.5274$

Table 5.ane Comparing a serial with a p-series (p = two)

$\mathrm{one\; thousand}$	$\mathrm{i}$	$2$	$3$	$4$	$5$	$6$	$7$	$\mathrm{eight}$
${\displaystyle \sum _{n=\mathrm{one}}^{\mathrm{one\; thousand}}\frac{\mathrm{ane}}{\mathrm{due\; north}-1\text{/}2}}$	$\mathrm{two}$	$\mathrm{ii.6667}$	$3.0667$	$\mathrm{iii.3524}$	$\mathrm{three.5746}$	$3.7564$	$\mathrm{iii.9103}$	$4.0436$
${\displaystyle \sum _{n=\mathrm{ane}}^k\frac{\mathrm{ane}}{n}}$	$1$	$1.5$	$\begin{array}{l}1.8333\hfill \end{array}$	$2.0933$	$2.2833$	$2.45$	$2.5929$	$2.7179$

Table v.2 Comparing a series with the harmonic series

Theorem 5.11

Comparison Test

1. Suppose there exists an integer $N$ such that $0\le a_{\mathrm{north}}\le b_{\mathrm{north}}$ for all $n\ge N.$ If ${\displaystyle \sum }_{n=1}^{\infty }{b}_n$ converges, then ${\displaystyle \sum }_{\mathrm{north}=1}^{\infty }{a}_{\mathrm{north}}$ converges.
2. Suppose there exists an integer $\mathrm{Northward}$ such that $a_n\ge b_n\ge 0$ for all $\mathrm{northward}\ge \mathrm{Northward}.$ If ${\displaystyle \sum }_{\mathrm{due\; north}=1}^{\infty }{b}_{\mathrm{northward}}$ diverges, then ${\displaystyle \sum }_{n=1}^{\infty }{a}_n$ diverges.

Proof

We show office i. The proof of office ii. is the contrapositive of role i. Let $\left\{S_{\mathrm{one\; thousand}}\right\}$ be the sequence of partial sums associated with ${\displaystyle \sum }_{\mathrm{due\; north}=\mathrm{ane}}^{\infty }{a}_{\mathrm{due\; north}},$ and permit $L={\displaystyle \sum }_{n=1}^{\infty }{b}_n.$ Since the terms $a_n\ge 0,$

$$S_{\mathrm{thousand}}=a_{\mathrm{one}}+a_{\mathrm{ii}}+\text{⋯}+a_{\mathrm{1000}}\le a_{\mathrm{i}}+a_{\mathrm{two}}+\text{⋯}+a_g+a_{\mathrm{thousand}+1}={\mathrm{South}}_{k+1}.$$

Therefore, the sequence of fractional sums is increasing. Farther, since $a_{\mathrm{north}}\le b_n$ for all $n\ge N,$ then

$${\displaystyle \sum _{n=N}^k{a}_{\mathrm{due\; north}}\le }{\displaystyle \sum _{n=\mathrm{Due\; north}}^k{b}_n\le }{\displaystyle \sum _{n=\mathrm{ane}}^{\infty }{b}_n=L.}$$

Therefore, for all $k\ge \mathrm{ane},$

$$S_k=\left(a_1+a_2+\text{⋯}+a_{N-1}\right)+{\displaystyle \sum _{n=N}^{\mathrm{one\; thousand}}{a}_n\le }\left(a_{\mathrm{ane}}+a_2+\text{⋯}+a_{\mathrm{North}-\mathrm{one}}\right)+\mathrm{50}.$$

Since $a_{\mathrm{i}}+a_{\mathrm{two}}+\text{⋯}+a_{N-1}$ is a finite number, we conclude that the sequence $\left\{{\mathrm{South}}_{\mathrm{1000}}\right\}$ is bounded above. Therefore, $\left\{S_m\right\}$ is an increasing sequence that is bounded higher up. Past the Monotone Convergence Theorem, nosotros conclude that $\left\{{\mathrm{Due\; south}}_g\right\}$ converges, and therefore the serial ${\displaystyle \sum _{\mathrm{due\; north}=1}^{\infty }{a}_n}$ converges.

□

To use the comparison test to determine the convergence or divergence of a series ${\displaystyle \sum _{n=1}^{\infty }{a}_n},$ information technology is necessary to find a suitable series with which to compare information technology. Since we know the convergence properties of geometric serial and p-serial, these series are often used. If in that location exists an integer $\mathrm{North}$ such that for all $n\ge \mathrm{Northward},$ each term $a_{\mathrm{northward}}$ is less than each corresponding term of a known convergent serial, then ${\displaystyle \sum _{n=1}^{\infty }{a}_{\mathrm{due\; north}}}$ converges. Similarly, if there exists an integer $\mathrm{North}$ such that for all $n\ge N,$ each term $a_{\mathrm{due\; north}}$ is greater than each corresponding term of a known divergent series, then ${\displaystyle \sum _{n=\mathrm{ane}}^{\infty }{a}_{\mathrm{northward}}}$ diverges.

Example 5.17

Using the Comparison Exam

For each of the following series, use the comparison examination to make up one's mind whether the series converges or diverges.

1. ${\displaystyle \sum }_{n=\mathrm{ane}}^{\infty }\frac{\mathrm{one}}{n^3+3n+\mathrm{i}}$
2. ${\displaystyle \sum }_{n=1}^{\infty }\frac{1}{2^n+\mathrm{one}}$
3. ${\displaystyle \sum }_{\mathrm{due\; north}=2}^{\infty }\frac{1}{\text{ln}\left(n\right)}$

Checkpoint 5.16

Employ the comparing exam to make up one's mind if the series ${\displaystyle \sum }_{n=1}^{\infty }\frac{n}{n^{\mathrm{three}}+\mathrm{due\; north}+1}$ converges or diverges.

Limit Comparison Test

The comparison test works nicely if we can find a comparable series satisfying the hypothesis of the test. Notwithstanding, sometimes finding an appropriate series can be difficult. Consider the series

$${\displaystyle \sum }_{n=\mathrm{ii}}^{\infty }\frac{1}{{\mathrm{northward}}^2-\mathrm{ane}}.$$

It is natural to compare this series with the convergent series

$${\displaystyle \sum }_{\mathrm{northward}=2}^{\infty }\frac{1}{{\mathrm{north}}^2}.$$

However, this serial does not satisfy the hypothesis necessary to use the comparison test because

$$\frac{\mathrm{i}}{{\mathrm{due\; north}}^{\mathrm{ii}}-1}>\frac{1}{n^{\mathrm{ii}}}$$

for all integers $n\ge 2.$ Although we could wait for a different series with which to compare ${\displaystyle \sum }_{n=2}^{\infty }1\text{/}(n^{\mathrm{ii}}-\mathrm{ane}),$ instead we show how nosotros can use the limit comparing test to compare

$${\displaystyle \sum }_{n=2}^{\infty }\frac{1}{{\mathrm{north}}^2-1}\text{and}{\displaystyle \sum }_{n=2}^{\infty }\frac{\mathrm{i}}{{\mathrm{due\; north}}^{\mathrm{ii}}}.$$

Let usa examine the idea backside the limit comparison examination. Consider 2 serial ${\displaystyle \sum }_{\mathrm{northward}=\mathrm{ane}}^{\infty }{a}_n$ and ${\displaystyle \sum }_{\mathrm{northward}=1}^{\infty }{b}_n.$ with positive terms $a_n\text{and}{b}_{\mathrm{northward}}$ and evaluate

$$\underset{\mathrm{due\; north}\to \infty }{\text{lim}}\frac{a_n}{b_n}.$$

If

$$\underset{n\to \infty }{\text{lim}}\frac{a_n}{b_{\mathrm{due\; north}}}=L\ne 0,$$

then, for $\mathrm{northward}$ sufficiently large, $a_{\mathrm{north}}\approx \mathrm{Fifty}{b}_n.$ Therefore, either both series converge or both series diverge. For the series ${\displaystyle \sum }_{\mathrm{due\; north}=\mathrm{two}}^{\infty }\mathrm{one}\text{/}(n^{\mathrm{ii}}-1)$ and ${\displaystyle \sum }_{n=2}^{\infty }1\text{/}{\mathrm{north}}^{\mathrm{two}},$ we see that

$$\underset{n\to \infty }{\text{lim}}\frac{1\text{/}(n^{\mathrm{ii}}-1)}{\mathrm{ane}\text{/}{n}^2}=\underset{n\to \infty }{\text{lim}}\frac{n^{\mathrm{ii}}}{{\mathrm{north}}^2-\mathrm{i}}=\mathrm{one}.$$

Since ${\displaystyle \sum }_{n=2}^{\infty }1\text{/}{n}^2$ converges, we conclude that

$${\displaystyle \sum }_{n=2}^{\infty }\frac{1}{{\mathrm{north}}^{\mathrm{two}}-\mathrm{ane}}$$

converges.

The limit comparison test tin can exist used in 2 other cases. Suppose

$$\underset{n\to \infty }{\text{lim}}\frac{a_n}{b_n}=0.$$

In this case, $\left\{a_{\mathrm{northward}}\text{/}{b}_{\mathrm{due\; north}}\right\}$ is a bounded sequence. As a outcome, at that place exists a constant $M$ such that $a_n\le G{b}_{\mathrm{north}}.$ Therefore, if ${\displaystyle \sum }_{n=\mathrm{i}}^{\infty }{b}_{\mathrm{due\; north}}$ converges, then ${\displaystyle \sum }_{n=1}^{\infty }{a}_n$ converges. On the other hand, suppose

$$\underset{\mathrm{northward}\to \infty }{\text{lim}}\frac{a_n}{b_n}=\infty .$$

In this case, $\left\{a_{\mathrm{due\; north}}\text{/}{b}_{\mathrm{north}}\right\}$ is an unbounded sequence. Therefore, for every constant $M$ there exists an integer $N$ such that $a_n\ge \mathrm{Thousand}{b}_{\mathrm{due\; north}}$ for all $n\ge N.$ Therefore, if ${\displaystyle \sum }_{\mathrm{north}=\mathrm{one}}^{\infty }{b}_n$ diverges, then ${\displaystyle \sum }_{\mathrm{northward}=1}^{\infty }{a}_{\mathrm{north}}$ diverges as well.

Theorem 5.12

Limit Comparison Test

Allow $a_n,b_{\mathrm{northward}}\ge 0$ for all $n\ge 1.$

1. If $\underset{\mathrm{due\; north}\to \infty }{\text{lim}}{a}_{\mathrm{north}}\text{/}{b}_n=\mathrm{50}\ne 0,$ then ${\displaystyle \sum }_{\mathrm{north}=\mathrm{ane}}^{\infty }{a}_n$ and ${\displaystyle \sum }_{n=\mathrm{one}}^{\infty }{b}_n$ both converge or both diverge.
2. If $\underset{\mathrm{north}\to \infty }{\text{lim}}{a}_n\text{/}{b}_n=0$ and ${\displaystyle \sum }_{n=\mathrm{one}}^{\infty }{b}_{\mathrm{northward}}$ converges, then ${\displaystyle \sum }_{\mathrm{north}=1}^{\infty }{a}_{\mathrm{north}}$ converges.
3. If $\underset{\mathrm{due\; north}\to \infty }{\text{lim}}{a}_n\text{/}{b}_{\mathrm{north}}=\infty$ and ${\displaystyle \sum }_{\mathrm{north}=1}^{\infty }{b}_n$ diverges, so ${\displaystyle \sum }_{\mathrm{due\; north}=1}^{\infty }{a}_{\mathrm{north}}$ diverges.

Note that if $a_{\mathrm{north}}\text{/}{b}_n\to 0$ and ${\displaystyle \sum }_{\mathrm{northward}=\mathrm{ane}}^{\infty }{b}_n$ diverges, the limit comparison test gives no data. Similarly, if $a_n\text{/}{b}_{\mathrm{northward}}\to \infty$ and ${\displaystyle \sum }_{n=1}^{\infty }{b}_n$ converges, the exam besides provides no information. For example, consider the two series ${\displaystyle \sum _{n=1}^{\infty }\mathrm{one}}\text{/}\sqrt{n}$ and ${\displaystyle \sum _{n=1}^{\infty }\mathrm{i}}\text{/}{n}^2.$ These serial are both p-serial with $p=\mathrm{ane}\text{/}2$ and $p=2,$ respectively. Since $p=\mathrm{ane}\text{/}2<\mathrm{ane},$ the series ${\displaystyle \sum _{n=1}^{\infty }\mathrm{one}}\text{/}\sqrt{\mathrm{due\; north}}$ diverges. On the other paw, since $p=\mathrm{ii}>1,$ the series ${\displaystyle \sum _{\mathrm{northward}=1}^{\infty }\mathrm{i}}\text{/}{\mathrm{north}}^{\mathrm{ii}}$ converges. However, suppose we attempted to apply the limit comparison examination, using the convergent $p-\text{series}$ ${\displaystyle \sum _{n=1}^{\infty }1}\text{/}{n}^3$ as our comparison series. Outset, we run into that

$$\frac{1\text{/}\sqrt{\mathrm{north}}}{1\text{/}{n}^3}=\frac{{\mathrm{northward}}^{\mathrm{iii}}}{\sqrt{n}}=n^{5\text{/}2}\to \infty \text{as}n\to \infty .$$

Similarly, we see that

$$\frac{\mathrm{i}\text{/}{\mathrm{north}}^2}{1\text{/}{n}^3}=n\to \infty \text{as}n\to \infty .$$

Therefore, if $a_n\text{/}{b}_n\to \infty$ when ${\displaystyle \sum _{n=1}^{\infty }{b}_n}$ converges, nosotros do not gain whatever information on the convergence or divergence of ${\displaystyle \sum _{n=1}^{\infty }{a}_n}.$

Example v.eighteen

Using the Limit Comparison Test

For each of the post-obit series, use the limit comparison test to make up one's mind whether the series converges or diverges. If the test does not utilize, say so.

1. ${\displaystyle \sum }_{n=1}^{\infty }\frac{\mathrm{i}}{\sqrt{\mathrm{northward}}+1}$
2. ${\displaystyle \sum }_{n=1}^{\infty }\frac{{\mathrm{ii}}^n+1}{3^{\mathrm{north}}}$
3. ${\displaystyle \sum }_{n=\mathrm{ane}}^{\infty }\frac{\text{ln}\left(n\right)}{{\mathrm{north}}^{\mathrm{ii}}}$

Checkpoint 5.17

Use the limit comparing test to determine whether the series ${\displaystyle \sum }_{\mathrm{north}=1}^{\infty }\frac{5^{\mathrm{northward}}}{3^{\mathrm{north}}+\mathrm{ii}}$ converges or diverges.

Section 5.4 Exercises

Apply the comparison test to determine whether the following serial converge.

194 .

${\displaystyle \sum }_{\mathrm{northward}=1}^{\infty }{a}_n$ where $a_n=\frac{2}{n\left(\mathrm{northward}+1\right)}$

195 .

${\displaystyle \sum }_{n=1}^{\infty }{a}_n$ where $a_n=\frac{1}{n\left(n+1\text{/}\mathrm{two}\right)}$

196 .

${\displaystyle \sum }_{n=\mathrm{one}}^{\infty }\frac{\mathrm{ane}}{2\left(n+\mathrm{one}\right)}$

197 .

${\displaystyle \sum }_{n=\mathrm{ane}}^{\infty }\frac{1}{2\mathrm{northward}-1}$

198 .

${\displaystyle \sum }_{n=2}^{\infty }\frac{1}{{\left(n\text{ln}n\right)}^2}$

199 .

${\displaystyle \sum }_{n=\mathrm{one}}^{\infty }\frac{n\text{!}}{\left(n+2\right)\text{!}}$

200 .

${\displaystyle \sum }_{n=\mathrm{one}}^{\infty }\frac{\mathrm{ane}}{\mathrm{due\; north}\text{!}}$

201 .

${\displaystyle \sum _{\mathrm{northward}=1}^{\infty }\frac{\text{sin}\left(1\text{/}\mathrm{north}\right)}{\mathrm{northward}}}$

202 .

${\displaystyle \sum _{\mathrm{north}=1}^{\infty }\frac{{\text{sin}}^{\mathrm{ii}}n}{{\mathrm{north}}^{\mathrm{ii}}}}$

203 .

${\displaystyle \sum _{\mathrm{due\; north}=\mathrm{ane}}^{\infty }\frac{\text{sin}\left(1\text{/}n\right)}{\sqrt{n}}}$

204 .

${\displaystyle \sum }_{\mathrm{northward}=\mathrm{ane}}^{\infty }\frac{n^{1.2}-1}{n^{\mathrm{2.three}}+1}$

205 .

${\displaystyle \sum }_{n=1}^{\infty }\frac{\sqrt{\mathrm{northward}+1}-\sqrt{\mathrm{north}}}{n}$

206 .

${\displaystyle \sum }_{\mathrm{due\; north}=1}^{\infty }\frac{\sqrt[4]{n}}{\sqrt[\mathrm{iii}]{n^4+n^{\mathrm{ii}}}}$

Utilise the limit comparison examination to determine whether each of the following series converges or diverges.

207 .

${\displaystyle \sum }_{\mathrm{due\; north}=\mathrm{one}}^{\infty }{\left(\frac{\text{ln}n}{\mathrm{north}}\right)}^{\mathrm{ii}}$

208 .

${\displaystyle \sum }_{n=1}^{\infty }{\left(\frac{\text{ln}n}{n^{0.6}}\right)}^2$

209 .

${\displaystyle \sum }_{n=1}^{\infty }\frac{\text{ln}\left(1+\frac{1}{\mathrm{northward}}\right)}{n}$

210 .

${\displaystyle \sum }_{n=1}^{\infty }\text{ln}\left(\mathrm{ane}+\frac{1}{n^{\mathrm{two}}}\right)$

211 .

${\displaystyle \sum }_{n=1}^{\infty }\frac{1}{4^{\mathrm{north}}-3^{\mathrm{due\; north}}}$

212 .

${\displaystyle \sum }_{n=1}^{\infty }\frac{\mathrm{ane}}{n^2-\mathrm{north}\text{sin}n}$

213 .

${\displaystyle \sum }_{\mathrm{north}=1}^{\infty }\frac{1}{e^{\left(\mathrm{i.1}\right)n}-3^n}$

214 .

${\displaystyle \sum }_{\mathrm{northward}=1}^{\infty }\frac{1}{e^{\left(1.01\right)n}-3^n}$

215 .

${\displaystyle \sum }_{n=1}^{\infty }\frac{1}{{\mathrm{due\; north}}^{1+1\text{/}\mathrm{northward}}}$

216 .

${\displaystyle \sum }_{\mathrm{due\; north}=1}^{\infty }\frac{\mathrm{i}}{{\mathrm{ii}}^{1+1\text{/}\mathrm{north}}{\mathrm{north}}^{1+1\text{/}\mathrm{north}}}$

217 .

${\displaystyle \sum }_{n=1}^{\infty }\left(\frac{1}{n}-\text{sin}\left(\frac{1}{\mathrm{due\; north}}\right)\right)$

218 .

${\displaystyle \sum }_{n=1}^{\infty }\left(\mathrm{i}-\text{cos}\left(\frac{1}{n}\right)\right)$

219 .

${\displaystyle \sum }_{n=1}^{\infty }\frac{1}{\mathrm{north}}(\frac{\pi }{2}-{\text{tan}}^{\mathrm{-ane}}\mathrm{north})$

220 .

${\displaystyle \sum }_{n=\mathrm{ane}}^{\infty }{\left(\mathrm{ane}-\frac{\mathrm{i}}{n}\right)}^{\mathrm{north}.n}$ (Hint: ${\left(1-\frac{1}{n}\right)}^{\mathrm{due\; north}}\to \mathrm{i}\text{/}e.)$

221 .

${\displaystyle \sum }_{\mathrm{north}=\mathrm{i}}^{\infty }\left(1-e^{\mathrm{-i}\text{/}n}\right)$ (Hint: $1\text{/}e\approx {\left(1-1\text{/}\mathrm{due\; north}\right)}^n,$ and then $1-e^{\mathrm{-1}\text{/}n}\approx 1\text{/}n.)$

222 .

Does ${\displaystyle \sum }_{n=\mathrm{two}}^{\infty }\frac{\mathrm{ane}}{{\left(\text{ln}n\right)}^p}$ converge if $p$ is large plenty? If so, for which $p\text{?}$

223 .

Does ${\displaystyle \sum }_{\mathrm{north}=1}^{\infty }{\left(\frac{\left(\text{ln}\mathrm{north}\right)}{\mathrm{northward}}\right)}^p$ converge if $p$ is large enough? If so, for which $p\text{?}$

224 .

For which $p$ does the serial ${\displaystyle \sum }_{\mathrm{northward}=1}^{\infty }{\mathrm{ii}}^{p\mathrm{north}}\text{/}{3}^{\mathrm{due\; north}}$ converge?

225 .

For which $p>0$ does the series ${\displaystyle \sum }_{n=1}^{\infty }\frac{{\mathrm{due\; north}}^p}{2^n}$ converge?

226 .

For which $r>0$ does the series ${\displaystyle \sum }_{n=1}^{\infty }\frac{r^{n^{\mathrm{two}}}}{{\mathrm{ii}}^n}$ converge?

227 .

For which $r>0$ does the serial ${\displaystyle \sum }_{n=1}^{\infty }\frac{2^n}{r^{n^2}}$ converge?

228 .

Find all values of $p$ and $q$ such that ${\displaystyle \sum }_{\mathrm{due\; north}=\mathrm{i}}^{\infty }\frac{n^p}{{\left(\mathrm{northward}\text{!}\right)}^q}$ converges.

229 .

Does ${\displaystyle \sum }_{n=\mathrm{i}}^{\infty }\frac{{\text{sin}}^2\left(nr\text{/}2\right)}{n}$ converge or diverge? Explain.

230 .

Explain why, for each $n,$ at least one of $\{\left|\text{sin}\mathrm{northward}\right|,\left|\text{sin}\left(\mathrm{northward}+1\right)\right|\text{,...},\left|\text{sin}n+6\right|\}$ is larger than $\mathrm{one}\text{/}\mathrm{two}.$ Use this relation to test convergence of ${\displaystyle \sum }_{\mathrm{northward}=1}^{\infty }\frac{\left|\text{sin}\mathrm{northward}\right|}{\sqrt{n}}.$

231 .

Suppose that $a_{\mathrm{due\; north}}\ge 0$ and $b_n\ge 0$ and that ${\displaystyle \sum _{\mathrm{north}=1}^{\infty }{a}^2{}_n}$ and ${\displaystyle \sum _{n=1}^{\infty }{b}^2{}_n}$ converge. Evidence that ${\displaystyle \sum _{n=1}^{\infty }{a}_{\mathrm{northward}}{b}_n}$ converges and ${\displaystyle \sum _{\mathrm{due\; north}=1}^{\infty }{a}_{\mathrm{due\; north}}}{b}_n\le \frac{\mathrm{i}}{2}\left({\displaystyle \sum _{n=1}^{\infty }{a}_{\mathrm{northward}}^2}+{\displaystyle \sum _{n=1}^{\infty }{b}_n^{\mathrm{ii}}}\right).$

232 .

Does ${\displaystyle \sum _{\mathrm{northward}=\mathrm{one}}^{\infty }{2}^{\text{−}\text{ln}\text{ln}n}}$ converge? (Hint: Write $2^{\text{ln}\text{ln}n}$ equally a power of $\text{ln}\mathrm{north}.)$

233 .

Does ${\displaystyle \sum _{n=\mathrm{one}}^{\infty }{\left(\text{ln}n\right)}^{\text{−}\text{ln}n}}$ converge? (Hint: Use $\mathrm{north}=e^{\text{ln}\left(n\right)}$ to compare to a $p-\text{serial}\text{.})$

234 .

Does ${\displaystyle \sum _{\mathrm{due\; north}=\mathrm{two}}^{\infty }{\left(\text{ln}n\right)}^{\text{−}\text{ln}\text{ln}n}}$ converge? (Hint: Compare $a_{\mathrm{due\; north}}$ to $1\text{/}n.)$

235 .

Testify that if $a_n\ge 0$ and ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ converges, and then ${\displaystyle \sum _{\mathrm{due\; north}=1}^{\infty }{a}^{\mathrm{two}}{}_{\mathrm{northward}}}$ converges. If ${\displaystyle \sum _{n=1}^{\infty }{a}^2{}_n}$ converges, does ${\displaystyle \sum _{\mathrm{due\; north}=\mathrm{i}}^{\infty }{a}_n}$ necessarily converge?

236 .

Suppose that $a_n>0$ for all $n$ and that ${\displaystyle \sum _{\mathrm{north}=1}^{\infty }{a}_{\mathrm{northward}}}$ converges. Suppose that $b_{\mathrm{due\; north}}$ is an arbitrary sequence of zeros and ones. Does ${\displaystyle \sum _{n=1}^{\infty }{a}_n}{b}_{\mathrm{north}}$ necessarily converge?

237 .

Suppose that $a_{\mathrm{north}}>0$ for all $n$ and that ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ diverges. Suppose that $b_n$ is an arbitrary sequence of zeros and ones with infinitely many terms equal to 1. Does ${\displaystyle \sum _{\mathrm{due\; north}=1}^{\infty }{a}_n}{b}_{\mathrm{north}}$ necessarily diverge?

238 .

Complete the details of the post-obit argument: If ${\displaystyle \sum _{\mathrm{northward}=\mathrm{one}}^{\infty }\frac{1}{n}}$ converges to a finite sum $s,$ then $\frac{1}{\mathrm{two}}\mathrm{due\; south}=\frac{\mathrm{i}}{2}+\frac{1}{4}+\frac{\mathrm{ane}}{\mathrm{vi}}+\text{⋯}$ and $s-\frac{1}{\mathrm{ii}}\mathrm{due\; south}=\mathrm{one}+\frac{1}{\mathrm{iii}}+\frac{\mathrm{ane}}{\mathrm{five}}+\text{⋯}.$ Why does this lead to a contradiction?

239 .

Show that if $a_n\ge 0$ and ${\displaystyle \sum _{\mathrm{north}=\mathrm{ane}}^{\infty }{a}^2{}_{\mathrm{northward}}}$ converges, then ${\displaystyle \sum _{n=1}^{\infty }{\text{sin}}^2\left(a_n\right)}$ converges.

240 .

Suppose that $a_n\text{/}{b}_{\mathrm{north}}\to 0$ in the comparison test, where $a_n\ge 0$ and $b_n\ge 0.$ Prove that if ${\displaystyle \sum b_n}$ converges, then ${\displaystyle \sum a_{\mathrm{due\; north}}}$ converges.

241 .

Allow $b_n$ be an infinite sequence of zeros and ones. What is the largest possible value of $\mathrm{ten}={\displaystyle \sum _{n=\mathrm{i}}^{\infty }{b}_{\mathrm{north}}\text{/}{2}^{\mathrm{due\; north}}}\text{?}$

242 .

Let $d_n$ be an space sequence of digits, significant $d_n$ takes values in $\{0,\mathrm{one}\text{,…},9\}.$ What is the largest possible value of $\mathrm{10}={\displaystyle \sum _{n=1}^{\infty }{d}_{\mathrm{due\; north}}\text{/}{\mathrm{ten}}^n}$ that converges?

243 .

Explicate why, if $x>1\text{/}\mathrm{ii},$ so $x$ cannot be written $x={\displaystyle \sum _{\mathrm{due\; north}=2}^{\infty }\frac{b_n}{2^{\mathrm{north}}}}\left(b_{\mathrm{due\; north}}=0\text{or}1,b_1=0\right).$

244 .

[T] Evelyn has a perfect balancing scale, an unlimited number of $1\text{-kg}$ weights, and i each of $1\text{/}2\text{-kg},\mathrm{one}\text{/}\mathrm{four}\text{-kg},\mathrm{one}\text{/}8\text{-kg},$ and and so on weights. She wishes to counterbalance a meteorite of unspecified origin to capricious precision. Assuming the scale is big plenty, can she do information technology? What does this have to practice with infinite series?

245 .

[T] Robert wants to know his body mass to arbitrary precision. He has a big balancing scale that works perfectly, an unlimited drove of $1\text{-kg}$ weights, and 9 each of $\mathrm{0.ane}\text{-kg,}$ $0.01\text{-kg},0.001\text{-kg,}$ and and then on weights. Bold the scale is large enough, can he practice this? What does this have to practise with infinite series?

246 .

The series ${\displaystyle \sum _{n=1}^{\infty }\frac{\mathrm{ane}}{\mathrm{ii}n}}$ is half the harmonic serial and hence diverges. Information technology is obtained from the harmonic series by deleting all terms in which $n$ is odd. Let $m>1$ be fixed. Show, more by and large, that deleting all terms $1\text{/}\mathrm{due\; north}$ where $n=mk$ for some integer $k$ also results in a divergent series.

247 .

In view of the previous practise, it may be surprising that a subseries of the harmonic series in which about ane in every five terms is deleted might converge. A depleted harmonic serial is a serial obtained from ${\displaystyle \sum _{n=\mathrm{one}}^{\infty }\frac{1}{n}}$ by removing any term $1\text{/}n$ if a given digit, say $9,$ appears in the decimal expansion of $n.$ Argue that this depleted harmonic series converges by answering the post-obit questions.

1. How many whole numbers $\mathrm{northward}$ take $d$ digits?
2. How many $d\text{-digit}$ whole numbers $h\left(d\right).$ practice not incorporate $9$ as one or more than of their digits?
3. What is the smallest $d\text{-digit}$ number $m\left(d\right)\text{?}$
4. Explain why the deleted harmonic series is divisional past ${\displaystyle \sum _{d=\mathrm{one}}^{\infty }\frac{h\left(d\right)}{\mathrm{grand}\left(d\right)}}.$
5. Show that ${\displaystyle \sum _{d=\mathrm{i}}^{\infty }\frac{h\left(d\right)}{\mathrm{grand}\left(d\right)}}$ converges.

248 .

Suppose that a sequence of numbers $a_n>0$ has the property that $a_1=1$ and $a_{n+1}=\frac{\mathrm{one}}{n+1}{S}_n,$ where ${\mathrm{Southward}}_n=a_{\mathrm{one}}+\text{⋯}+a_n.$ Tin can you determine whether ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ converges? (Hint: ${\mathrm{Southward}}_n$ is monotone.)

249 .

Suppose that a sequence of numbers $a_n>0$ has the belongings that $a_1=\mathrm{one}$ and $a_{\mathrm{north}+\mathrm{one}}=\frac{1}{{\left(\mathrm{north}+1\right)}^2}{\mathrm{Southward}}_n,$ where $S_n=a_{\mathrm{ane}}+\text{⋯}+a_{\mathrm{north}}.$ Can yous determine whether ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ converges? (Hint: $S_2=a_2+a_1=a_2+{\mathrm{South}}_{\mathrm{one}}=a_{\mathrm{two}}+1=1+1\text{/}4=\left(1+1\text{/}4\right)S_1,$ $S_3=\frac{\mathrm{one}}{3^2}{\mathrm{South}}_{\mathrm{two}}+S_2=\left(1+\mathrm{one}\text{/}9\right){\mathrm{Due\; south}}_{\mathrm{ii}}=\left(\mathrm{ane}+\mathrm{ane}\text{/}\mathrm{ix}\right)\left(1+1\text{/}4\right){\mathrm{Southward}}_1,$ etc. Look at $\text{ln}\left({\mathrm{Due\; south}}_{\mathrm{north}}\right),$ and apply $\text{ln}\left(1+t\right)\le t,$ $t>0.)$

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Source: https://openstax.org/books/calculus-volume-2/pages/5-4-comparison-tests

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