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}

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

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