A necessary prerequisite for the convergence of an infinite series involves the…

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

A necessary prerequisite for the convergence of an infinite series involves the long–run behavior of its terms. Consider the sequence of terms

(a_n)

for an infinite series

\displaystyle\sum_{n=1}^{\infty} a_n

. Answer the following questions to demonstrate conceptual understanding of this prerequisite condition. (a) State the condition that the sequence

(a_n)

must satisfy in order for

\displaystyle\sum_{n=1}^{\infty} a_n

to have any chance of converging. (b) Explain, without performing any tests for convergence beyond part (a), why the series

\displaystyle\sum_{n=1}^{\infty} \dfrac{3n+2}{n}

cannot converge. (c) The terms of the series

\displaystyle\sum_{n=2}^{\infty} \dfrac{1}{n\ln n}

approach

0

n\to\infty

. Does the condition from part (a) guarantee that this series converges? Provide a brief explanation.