Which statement best explains why the improper integral of from to cannot be used to…

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

Problem

Which statement best explains why the improper integral of

f(x)=\dfrac{(-1)^{x}}{x^{2}}

from

1

\infty

cannot be used to decide the convergence of the series

\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{2}}

? (a) The Integral Test requires the function to be positive, and because

f(x)

changes sign the test cannot be used. (b)

f(x)

is not defined (as a real-valued function) for non-integer

x

, so the improper integral

\displaystyle\int_{1}^{\infty}f(x)\,dx

is not meaningful. (c) Because the alternating signs would cause the integral to cancel out and converge to

0

, even if the series diverged. (d) Since the series already converges by another test, checking an improper integral would be redundant.