The series is telescoping if we can cancel all of the terms in the middle (every term but the first and last). ... Since s exists as a real number, the sum of the series is s = 1 s=1 s=1, and we can conclude that the series of partial sums s n s_n sn​ converges, and therefore that the series a n a_n an​ also converges.

Does a telescoping series converge or diverge?

because of cancellation of adjacent terms. So, the sum of the series, which is the limit of the partial sums, is 1. and any infinite sum with a constant term diverges.

What makes a series telescoping?

Telescoping series is a series where all terms cancel out except for the first and last one. This makes such series easy to analyze.