Alternating Series Test — Question 8

*Question 8 Consider $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n}{\ln(n+1)}$.

1. Verify decrease without differentiating.

2. Apply the AST.

3. Test absolute convergence and classify the series.

*Question 8 – Solution

Since $n+2>n+1$ and logarithm is strictly increasing, $$\ln(n+2)>\ln(n+1)>0.$$ Taking positive reciprocals reverses the inequality, so $b_{n+1}<b_n$ for $b_n=1/\ln(n+1)$.

Because $\ln(n+1)\to\infty$, $b_n\to0$. Thus the alternating series converges.

Since $\ln(n+1)\le n$ for $n\ge1$, $$\frac1{\ln(n+1)}\ge\frac1n.$$ The absolute series diverges by comparison with the harmonic series. Therefore convergence is conditional.