Answer
a) $\sum_{n=1}^{\infty} b_n$
b) 1) $b_{n+1}\leq b_n$ for all $n$
2) $\displaystyle{\lim_{n \to \infty}} b_n=0$
c) $|R_n|\leq b_{n+1}$
Work Step by Step
a) An alternating series is a series which has the terms alternating in sign: any consecutive terms have opposite signs.
b) An alternating series converges when the following two conditions are met:
1) each term is greater than the previous one
2) the limit of the terms sequence is zero.
This can be written like this:
The alternating series $\sum_{n=1}^{\infty} (-1)^{n-1}b_n$ is convergent if it satisfies:
1) $b_{n+1}\leq b_n$ for all $n$
2) $\displaystyle{\lim_{n \to \infty}} b_n=0$.
c) If a series $S=\sum_{n=1}^{\infty} (-1)^{n-1}b_n$ is convergent, the rest $R_n$ satisfies:
$|R_n|=|S-S_n|\leq b_{n+1}$
