Let . It can be shown that . Which of the following statements about is…

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

Problem

Let

T=\displaystyle\sum_{n=2}^{\infty}\frac{\ln n}{n^{2}}

. It can be shown that

\displaystyle\int_{2}^{\infty}\frac{\ln x}{x^{2}}\,dx=\frac{1+\ln 2}{2}

. Which of the following statements about

T

is true? (a)

\displaystyle T=\frac{1+\ln 2}{2}

(b)

\displaystyle\frac{\ln 2}{4}<T<\frac{1+\ln 2}{2}

(c)

\displaystyle\frac{1+\ln 2}{2}<T<\frac{1+\ln 2}{2}+\frac{\ln 2}{4}

(d)

\displaystyle\frac{1+\ln 2}{2}<T<\frac{1+\ln 2}{2}+\frac{\ln 2}{2}