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Chapter 7: Problem 58

Use the Limit Comparison Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{n}{(n+1) 2^{n-1}} $$

Short Answer

Expert verified
The original series is convergent.

Step by step solution

01

Recognize the given series

First, acknowledge the series given to us is: \(\sum_{n=1}^{\infty} \frac{n}{(n+1) 2^{n-1}}\)
02

Choose a suitable comparison series

Now, choose a series to compare the given one with. For this problem, a good choice would be the convergent geometric series \(\sum_{n=1}^{\infty} \frac{1}{2^n}\)
03

Apply the Limit Comparison Test

Now that a comparison series has been chosen, apply the Limit Comparison Test which states that if you have two series \(\sum_{n=1}^{\infty} a_n\) and \(\sum_{n=1}^{\infty} b_n\) where a sequence \(\{b_n\}\) is positive and \(\lim_{n \to \infty} \frac{a_n}{b_n} = L\) (where L is finite and positive), both series will either converge or diverge. Compute the limit as n approaches infinity for the ratio of the given series to the comparison series: \( \lim_{n \to \infty} \frac{(n/(n+1) 2^{n-1})}{(1/2^n)} \).
04

Simplify the limit

Simplify the limit equation to obtain, \( \lim_{n \to \infty} \frac{n}{n+1} = 1 \).
05

State the final conclusion

Given that \(\sum_{n=1}^{\infty} \frac{1}{2^n}\) is a convergent series and our limit is finite and positive, by the Limit Comparison Test, we can conclude that the original series \(\sum_{n=1}^{\infty} \frac{n}{(n+1) 2^{n-1}}\) is also convergent.

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Most popular questions from this chapter

Use a graphing utility to graph the function. Identify the horizontal asymptote of the graph and determine its relationship to the sum of the series. $$ \frac{\text { Function }}{f(x)=2\left[\frac{1-(0.8)^{x}}{1-0.8}\right]} \frac{\text { Series }}{\sum_{n=0}^{\infty} 2\left(\frac{4}{5}\right)^{n}} $$

Find all values of \(x\) for which the series converges. For these values of \(x,\) write the sum of the series as a function of \(x\). $$ \sum_{n=0}^{\infty} 4\left(\frac{x-3}{4}\right)^{n} $$

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An electronic games manufacturer producing a new product estimates the annual sales to be 8000 units. Each year, \(10 \%\) of the units that have been sold will become inoperative. So, 8000 units will be in use after 1 year, \([8000+0.9(8000)]\) units will be in use after 2 years, and so on. How many units will be in use after \(n\) years?
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