A student wishes to analyze the convergence of the series by evaluating an improper…

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

A student wishes to analyze the convergence of the series

\displaystyle \sum_{n=1}^{\infty} (-1)^{n-1}\dfrac{\sqrt{n}}{n+1}

by evaluating an improper integral. Which of the following approaches is valid? (a) Integrate

(-1)^{x-1}\dfrac{\sqrt{x}}{x+1}

from

1

\infty

using the Integral Test. (b) Integrate

\dfrac{\sqrt{x}}{x+1}

from

1

\infty

and compare the result with the series of absolute values. (c) Integrate

\dfrac{\sqrt{x}}{x+1}

from

1

\infty

; since the integral diverges, conclude that the alternating series diverges as well. (d) Replace

\dfrac{\sqrt{x}}{x+1}

\dfrac1{\sqrt{x}}

and apply the

p

–integral test, asserting convergence because

p=\tfrac12>1