Question

Please help
Find the sum. Is this convergent or divergent? Explain your answer.

Answer 1

A sum of a finite number of numbers must be convergent.

Consider the $k$-th partial sum for the series:

$S_k=\displaystyle\sum_{n=2}^k(-3)\cdot4^{n-1}$
$S_k=-3(4^1+4^2+\cdots+4^{k-2}+4^{k-1})$
$4S_k=-3(4^2+4^3+\cdots+4^{k-1}+4^k)$
$\implies S_k-4S_k=-3(4^1-4^k)$
$\implies S_k=4(1-4^{k-1})$

In this case, we have $k=5$, so the sum we want to find is

$S_5=4(1-4^{5-1})=-1020$