Does the series converge or diverge?

Question

Answer 1

The series can be rewritten as ∑_{(n = 2 to ∞)} [5(-3/7)ⁿ + 2(5/7)ⁿ] =

5∑(-3/7)ⁿ + 2∑(5/7)ⁿ

The first series is the tail end of a geometric with common ratio -3/7 and the second is the tail end of a geometric series with common ratio 5/7. Since both common ratios are between -1 and 1, both series converge, so the original series converges as well.