Prove that if {$a_n$} and {$b_n$} are sequences of complex numbers such that $a_n$ = $b_{n+1}$ - $b_n$ for any $n$ $\in$ $\Bbb Z$$^+$ , then $\sum_{n=1}^\infty a_n$ converges iff $\lim_ {n\to \infty} b_n$ exists, in which case we have
$\sum_{n=1}^\infty a_n$ = $\lim_{n\to \infty} b_n - b_1$
I'm so lost on this. How do I approach it? Any tips are greatly appreciated.
Hint: Consider the partial sums $S_N=\sum\limits_{n=1}^{N}a_n$. By definition, $\sum\limits_{n=1}^{N}a_n$ converges if and only if $S_N$ converges (i.e., $\lim\limits_{N\to\infty} S_N$ exists and is finite).
Can you see these are telescoping, and so can you write down explicitly what $S_N$ is?