A sequence of positive real numbers is known to be monotonically decreasing and satisfies

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

Problem

A sequence

(a_n)

of positive real numbers is known to be monotonically decreasing and satisfies

\displaystyle\lim_{n\to\infty}a_n=0

. Consider the infinite series

\displaystyle \sum_{n=1}^{\infty}(-1)^{n}a_n

. (a) Describe the specific properties of the sequence

(a_n)

that guarantee the series converges. In your explanation, indicate why each property is necessary. (b) Explain the practical meaning, in words, of the inequality

|S-S_n|\le a_{n+1}

, where

S

denotes the exact value of the series and

S_n

denotes its

n

th partial sum. (c) Suppose

a_n=\dfrac{1}{n^p}

for some constant

p>0

. Compare the convergence of

\displaystyle\sum_{n=1}^{\infty}(-1)^{n}a_n

with the convergence of

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

. Explain why the alternating version converges for every positive

p

, yet the non-alternating version converges only when

p>1

. Include clear reasoning based on the properties you stated in part (a).

Related theory

Browse the problem library