Question

How do you do the Alternating Series Test on this series and what is the result? sum_{n=2}^inftyfrac{(-1)^n}{n+1}

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asked 2021-02-15
How do you do the Alternating Series Test on this series and what is the result?
\(\sum_{n=2}^\infty\frac{(-1)^n}{n+1}\)

Answers (1)

2021-02-16
Alternating series:
A series of the form
\(\sum_{n=1}^\infty(-1)^{n+1}na_n=a_0-a_1+a_2-a_3+...\)
where either all \(a_n\) are positive or all \(a_n\) are negative, is called an alternating series.
The alternating series test then says: if \(|a_n|\) decreases monotonically and \(\lim_{n\rightarrow\infty}a_n=0\) then the alternating series converges.
Moreover, let L denote the sum of the series, then the partial sum
\(S_k=\sum_{n=0}^k(-1)^na_n\)
approximates L with error bounded by the next omitted term:
\(|S_k-L|\leq|S_k-S_{k+1}|=a_{k+1}\)
\(a_n=\frac{1}{n+1}\)
\(a_{n+1}=\frac{1}{n+2}\)
Therefore \(a_{n+1}
So \(a_n\) is monotonically decrea \(\sin g\)
\(a_n>0\) for all \(n=1,2,3,...\)
\(n\to\infty\)
\(=0\)
So the series converges.
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