Alternating Series Test — Question 5

*Question 5 Consider $\displaystyle\sum_{n=2}^{\infty}(-1)^n\frac{\ln n}{n}$.

1. Use a continuous derivative to prove eventual decrease of the magnitude.

2. Verify the zero-limit condition and apply the AST.

3. Test absolute convergence and classify the series.

*Question 5 – Solution

Let $f(x)=\ln x/x$. Then $$f'(x)=\frac{1-\ln x}{x^2}<0\qquad(x>e).$$ Hence $b_n=\ln n/n$ decreases for integers $n\ge3$. Eventual decrease is sufficient; finitely many initial terms do not affect convergence.

L’Hopital’s Rule gives $$\lim_{x\to\infty}\frac{\ln x}{x}=\lim_{x\to\infty}\frac1x=0.$$ Therefore the AST proves convergence.

For $n\ge3$, $\ln n\ge1$, so $\ln n/n\ge1/n$. The absolute-value series diverges by comparison with the harmonic series. Thus convergence is conditional.