Question

Describe various convergence tools for proving a sequence is convergent. (B) Prove the sequence ((x_n)) satisfying (x_n = 1 - 1n)

a) True
b) False

Answer 1

The sequence (x_n = 1 - 1/n) is convergent as n approaches infinity, since its limit evaluates to 1. Therefore, the statement is true.

Step-by-step explanation:

In mathematics, specifically in the study of sequences and series, there are several convergence tools that can be used to determine whether a sequence is convergent. These tools include, but are not limited to, the Nth term test for divergence, the comparison test, the ratio test, the root test, and the integral test. Each has certain conditions under which they can be appropriately applied to infer the convergence of a sequence.

Concerning the sequence (xn) with xn = 1 - 1/n, to prove its convergence, one typically examines the limit of the sequence as n approaches infinity.

We calculate the limit: lim (n->∞) (1 - 1/n) = lim (n->∞) 1 - lim (n->∞) 1/n = 1 - 0 = 1. Since the limit exists and is a finite number, the sequence (xn) is convergent and converges to 1.

This statement is true.