Alternating Series Test — Question 8

Question 8 – Solution

Step 1: Verify monotonicity.

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 $b_n=1/\ln(n+1)$.

Step 2: Check the limit and apply AST.

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

Step 3: Test absolute convergence.

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.