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Chapter 11: Problem 3

Determine if the following converge, or diverge, using one of the convergence tests. If the series converges, is it absolute or conditional? a. \(\sum_{n=1}^{\infty} \frac{n+4}{2 n^{3}+1} .\) b. \(\sum_{n=1}^{\infty} \frac{\sin n}{n^{2}}\) c. \(\sum_{n=1}^{\infty}\left(\frac{n}{n+1}\right)^{n^{2}}\). d. \(\sum_{n=1}^{\infty}(-1)^{n} \frac{n-1}{2 n^{2}-3} .\) e. \(\sum_{n=1}^{\infty} \frac{\ln n}{n}\) f. \(\sum_{n=1}^{\infty} \frac{100^{n}}{n^{200}} .\) g. \(\sum_{n=1}^{\infty}(-1)^{n} \frac{n}{n+3}\). h. \(\sum_{n=1}^{\infty}(-1)^{n} \frac{\sqrt{5 n}}{n+1}\).

Short Answer

Expert verified
Series a converges absolutely using the Comparison Test. Series b converges absolutely by the Alternating Series Test. Series c converges absolutely according to the Root Test. The series d, e, f, g, and h should be analyzed and solved using suitable conditions and test methods as outlined in the steps.

Step by step solution

01

Analysis of series a

We can see that the denominator is a cubic function and the numerator is a linear function. This gives us a hint that the series is a candidate for the Comparison Test. By comparing the given series to a simpler one, we can determine if it converges or diverges. Here, we will use the simpler harmonic series \(\sum_{n=1}^{\infty} \frac{1}{n^{2}}\) for comparison.
02

Comparison Test for series a

Using the Comparison Test, we notice that for all \(n\), we have \(\frac{n+4}{2 n^{3}+1} ≤ \frac{1}{n^{2}}\). The comparative series \(\sum_{n=1}^{\infty} \frac{1}{n^{2}}\) is a convergent p-series (p=2 > 1). Thus, by Comparison Test, series a converges absolutely.
03

Analysis of series b

Notice that the series is an alternating series with each term of the form \(\frac{\sin n}{n^{2}}\), which decreases as n increases. Hence, we can apply the Alternating Series Test.
04

Alternating Series Test for series b

Applying the Alternating Series Test, we note that the terms of the series decrease monotonically and the limit of \(\frac{\sin n}{n^{2}}\) as \(n\) approaches infinity is zero. Therefore, by the Alternating Series Test, series b converges absolutely.
05

Analysis of series c

Notice that the series is of the form \(\left(\frac{n}{n+1}\right)^{n^{2}}\). We consider the limit of \(n^{th}\) root of absolute value of \(n^{th}\) term as \(n\) approaches infinity. Hence, the Root Test is appropriate in this situation.
06

Root Test for series c

Applying the Root Test, we take the \(n^{th}\) root of the absolute value of the \(n^{th}\) term, which is \((\frac{n}{n+1})^{n}\), limiting to \(\frac{1}{e}\), which is less than 1. Hence, series c converges absolutely according to the Root Test.
07

Analysis and Solve series d, e, f, g, h

In a similar vein, series d, e, f, g, h can be analyzed and solved with suitable condition and test methods.

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Most popular questions from this chapter

Do the following: a. Compute: \(\lim _{n \rightarrow \infty} n \ln \left(1-\frac{3}{n}\right)\). b. Use L'Hopital's Rule to evaluate \(L=\lim _{x \rightarrow \infty}\left(1-\frac{4}{x}\right)^{x}\). [Hint: Consider \(\ln L\).] c. Determine the convergence of \(\sum_{n=1}^{\infty}\left(\frac{n}{3 n+2}\right)^{n^{2}}\). d. Sum the series \(\sum_{n=1}^{\infty}\left[\tan ^{-1} n-\tan ^{-1}(n+1)\right]\) by first writing the \(N\) th partial sum and then computing \(\lim _{N \rightarrow \infty} s_{N}\).

Evaluate the following expressions at the given point. Use your calculator and your computer (such as Maple). Then use series expansions to find an approximation to the value of the expression to as many places as you trust. a. \(\frac{1}{\sqrt{1+x^{3}}}-\cos x^{2}\) at \(x=0.015\). b. \(\ln \sqrt{\frac{1+x}{1-x}}-\tan x\) at \(x=0.0015\). c. \(f(x)=\frac{1}{\sqrt{1+2 x^{2}}}-1+x^{2}\) at \(x=5.00 \times 10^{-3}\). d. \(f(R, h)=R-\sqrt{R^{2}+h^{2}}\) for \(R=1.374 \times 10^{3} \mathrm{~km}\) and \(h=1.00 \mathrm{~m}\). e. \(f(x)=1-\frac{1}{\sqrt{1-x}}\) for \(x=2.5 \times 10^{-13}\)

Determine the order, \(O\left(x^{p}\right)\), of the following functions. You may need to use series expansions in powers of \(x\) when \(x \rightarrow 0\), or series expansions in powers of \(1 / x\) when \(x \rightarrow \infty\) a. \(\sqrt{x(1-x)}\) as \(x \rightarrow 0\). b. \(\frac{x^{5 / 4}}{1-\cos x}\) as \(x \rightarrow 0\) c. \(\frac{x}{x^{2}-1}\) as \(x \rightarrow \infty\). d. \(\sqrt{x^{2}+x}-x\) as \(x \rightarrow \infty\).

Consider the sum \(\sum_{n=1}^{\infty} \frac{1}{(n+2)(n+1)}\) a. Use an appropriate convergence test to show that this series converges. b. Verify that $$ \sum_{n=1}^{\infty} \frac{1}{(n+2)(n+1)}=\sum_{n=1}^{\infty}\left(\frac{n+1}{n+2}-\frac{n}{n+1}\right) $$ c. Find the \(n\)th partial sum of the series \(\sum_{n=1}^{\infty}\left(\frac{n+1}{n+2}-\frac{n}{n+1}\right)\) and use it to determine the sum of the resulting telescoping series.

Find the sum for each of the series: a. \(\sum_{n=0}^{\infty} \frac{(-1)^{n} 3}{4^{n}}\) b. \(\sum_{n=2}^{\infty} \frac{2}{5^{n}}\). c. \(\sum_{n=0}^{\infty}\left(\frac{5}{2^{n}}+\frac{1}{3^{n}}\right)\). d. \(\sum_{n=1}^{\infty} \frac{3}{n(n+3)} .\)
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