# Odd sequence in alternating series test proof

The author of this article (click here for image), when proving the alternating series test, computes the limit of the odd sequence as follows:

$$\lim_{n\rightarrow\infty}s_{2n+1}=\lim_{n\rightarrow\infty}\left(s_{2n}+b_{2n+1}\right)$$

$$\dots$$

How is $s_{2n+1}=s_{2n}+b_{2n+1}$?

The definition of the notation $s_n$ at that reference is $$s_n=b_1-b_2+b_3+\cdots+(-1)^{n-1}b_n$$ Therefore \begin{align*} s_{2n}&=b_1-b_2+b_3+\cdots-b_{2n}\\ s_{2n+1}&=b_1-b_2+b_3+\cdots-b_{2n}+b_{2n+1} \end{align*} so that $$s_{2n+1}=s_{2n}+b_{2n+1}$$
• Isn't $s_{2n}=s_{2n−2} +b_{2n-1}-b_{2n}$? – Jane Smith Jul 11 '15 at 4:43
By definition, $s_{k}$ is the sum of the first $k$ terms in the sequence: $$s_{2n+1} = \underbrace{ \ b_1 - b_2 + \dots + (-1)^{2n+1} b_{2n} \ }_{= \ s_{2n}} + \underbrace{ \ (-1)^{2n+2} b_{2n+1} \ }_{= \ b_{2n+1}}$$