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`