Absolute Convergence and Divergence — Question 6

Question 6 – Solution

Step 1: Verify the AST.

Let $b_n=1/(n\ln n)$. For $n\ge2$ it is positive. Its denominator is increasing, so $b_n$ decreases, and $b_n\to0$. Thus the alternating series converges.

Step 2: Test absolute convergence.

The absolute series is $\sum1/(n\ln n)$. For $f(x)=1/(x\ln x)$, $$\int_2^b f(x)\,dx=\ln(\ln b)-\ln(\ln2)\to\infty.$$ The Integral Test proves divergence.

Step 3: Classify and estimate.

The original series is conditionally convergent, with $$|S-s_N|\le\frac1{(N+1)\ln(N+1)}.$$