Mathematics

Problem

Define the sequence

\{a_n\}

a_n=\begin{cases} 1, & \text{if } n \text{ is an odd integer},\\ -1, & \text{if } n \text{ is an even integer}.\end{cases}

Consider the infinite series

\displaystyle \sum_{n=1}^{\infty} a_n

. (a) Compute explicit formulas for the even-indexed partial sums

S_{2k}

and the odd-indexed partial sums

S_{2k+1}

. (b) Determine the limits of the two subsequences

\{S_{2k}\}

and

\{S_{2k+1}\}

, if they exist. (c) Use the results from part (b) to decide whether the full series converges or diverges, and provide a clear justification.