Integral Test — Question 8

*Question 8 Consider $$\sum_{n=2}^{\infty}\frac{\ln n}{n^2}.$$

1. Verify the Integral Test hypotheses for $f(x)=\ln x/x^2$ on $[2,\infty)$.

2. Evaluate the improper integral using integration by parts.

3. Determine convergence and derive two-sided bounds for $R_N$.

4. Explain how the result illustrates polynomial growth dominating logarithmic growth.

*Question 8 – Solution

Let $f(x)=\ln x/x^2$ for $x\ge2$. It is continuous and positive. Also $$f'(x)=\frac{1-2\ln x}{x^3}<0\qquad(x\ge2),$$ so it decreases.

Take $u=\ln x$ and $dv=x^{-2}dx$. Then $du=dx/x$ and $v=-1/x$: $$\int\frac{\ln x}{x^2}\,dx=-\frac{\ln x}{x}-\frac1x+C.$$ Hence $$\int_2^{\infty}\frac{\ln x}{x^2}\,dx=\frac{\ln2+1}{2}<\infty,$$ and the series converges.

Since the tail integral from $A$ equals $(\ln A+1)/A$, $$\boxed{\frac{\ln(N+1)+1}{N+1}\le R_N\le\frac{\ln N+1}{N}}.$$ Quadratic denominator growth dominates the logarithmic numerator.