Using integral estimation to show that $ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$

Question

Show with Integral estimation that $$ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$$

$$f(k)=\frac {\ln k}{k^2} $$ For the integral it is : 1

But the other part is the estimation, the lower and upper bound.

I try to set $1+f(1)$ at the upper bound and $1+f(\infty)$ at the lower. The answer is incorrect. What to do?

Answer 1

HINT 1

The inequality can be proved using the general relation $$\int_N^{\infty}f(x)\,dx\leq \sum_{k=N}^{\infty}f(k)\leq f(N)+\int_N^{\infty}f(x)\,dx$$ that holds when the series converges (namely, the integral is finite).

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HINT 2

Also, note that in this case we have that $f(1)=0$, so $$\sum_{k=1}^{\infty}\frac{\ln(k)}{k^2}=\sum_{k=2}^{\infty}\frac{\ln(k)}{k^2}$$

What you need to show, as said in comments, is that $$\int_2^{\infty}f(x)\,dx\leq \sum_{k=2}^{\infty}\frac{\ln(k)}{k^2}\leq f(2)+\int_2^{\infty}f(x)\,dx$$

that is the relation of HINT 1 using the fact that $N=2$, because is the initial term of the series.

Try to use this relation to prove the inequality. Good luck!