Mathematics

Problem

Consider the series

\displaystyle\sum_{n=2}^{\infty}a_n

where

a_n=\dfrac{\ln n}{n^{2}}

. Define

f(x)=\dfrac{\ln x}{x^{2}}

for

x\ge 2

. a. Show that

f

is positive, continuous, and decreasing on the interval

[2,\infty)

. Provide algebraic or calculus–based reasoning for each property. b. Explain how the properties established in part (a) affect the convergence or divergence of the series

\displaystyle\sum_{n=2}^{\infty}a_n

. State your conclusion about the series and justify it thoroughly.

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