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Chapter 7: Problem 48

Find the sum of the convergent series by using a well-known function. Identify the function and explain how you obtained the sum. $$ \sum_{n=0}^{\infty}(-1)^{n} \frac{1}{2 n+1} $$

Short Answer

Expert verified
The sum of the series \(\sum_{n=0}^{\infty}(-1)^{n} \frac{1}{2 n+1}\) is \(\frac{\pi}{4}\) or approximately 0.785.

Step by step solution

01

Identify the series and function type

The given series \(\sum_{n=0}^{\infty}(-1)^{n} \frac{1}{2 n+1}\) is a form of an alternating series where the terms alternate in sign from positive to negative. The function form is \((-1)^{n} \frac{1}{2 n+1}\) where the denominator changes with each term \(n\).
02

Identify a similar well-known function

The well-known function this series is similar to is \(\arctan(x) = \sum_{n=0}^{\infty} (-1)^{n} \frac{x^{2n+1}}{2n+1}\). We can see the similarity as the general formulas for both series match.
03

Calculate the sum of the series

Since \((-1)^{n} \frac{x^{2n+1}}{2n+1}\) evaluated at \(x = 1\) is \((-1)^{n} \frac{1}{2 n+1}\), the sum of the series will be \(\arctan(1)\). The inverse tangent of 1 is \(\frac{\pi}{4}\), which is approximately 0.785.

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Most popular questions from this chapter

In Exercises \(47-52,\) (a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers $$ 0 . \overline{4} $$

Conjecture Let \(x_{0}=1\) and consider the sequence \(x_{n}\) given by the formula \(x_{n}=\frac{1}{2} x_{n-1}+\frac{1}{x_{n-1}}, \quad n=1,2, \ldots .\) Use a graphing utility to compute the first 10 terms of the sequence and make a conjecture about the limit of the sequence.

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The series \(\sum_{n=1}^{\infty} \frac{n}{1000(n+1)}\) diverges.

Consider the formula \(\frac{1}{x-1}=1+x+x^{2}+x^{3}+\cdots\) Given \(x=-1\) and \(x=2\), can you conclude that either of the following statements is true? Explain your reasoning. (a) \(\frac{1}{2}=1-1+1-1+\cdots\) (b) \(-1=1+2+4+8+\cdots\)

In Exercises 87 and 88 , 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. $$ \frac{\text { Function }}{f(x)=3\left[\frac{1-(0.5)^{x}}{1-0.5}\right]} \frac{\text { Series }}{\sum_{n=0}^{\infty} 3\left(\frac{1}{2}\right)^{n}} $$
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