Question

Whi is it a ncessary condition that the limit of the sequence approaches 0 for the limit of the seuqence of partial sums to exist?

A) To ensure convergence and prevent divergence.
B) To guarantee that the series is absolutely convergent.
C) To establish that the series converges conditionally.
D) To simplify calculations in the limit process.

Answer 1

A sequence must approach 0 for its series of partial sums to converge because if the sequence terms do not diminish, the sum will continue to grow infinitely and the series will diverge. Therefore correct option is A

Step-by-step explanation:

The question asks why it is a necessary condition that the limit of a sequence approaches 0 for the limit of the sequence of partial sums to exist. The reason for this requirement is to ensure convergence of the series. If the terms of a sequence do not approach 0, the partial sums will continue to grow larger and larger, and as a result, the series will diverge, meaning it does not have a finite limit.

Therefore, the correct answer is A) To ensure convergence and prevent divergence. This is a fundamental concept in the study of infinite series within calculus, which deals with the convergence criteria for series based on their sequence of terms.