Absolute Convergence and Divergence — Question 2

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

1. Prove ordinary convergence.

2. Test absolute convergence separately.

3. Classify the series, state its sum, and give an alternating error bound.

*Question 2 – Solution

For $b_n=1/n$, we have $b_n>0$, $b_n$ decreases, and $b_n\to0$. The Alternating Series Test therefore proves convergence.

$$\sum\left|\frac{(-1)^n}{n}\right|=\sum\frac1n,$$ which diverges. Thus the original series is conditionally convergent.

This sign convention is the negative alternating harmonic series, so $$S=-\ln2,\qquad |S-s_N|\le\frac1{N+1}.$$ Absolute convergence fails because removing the cancellation exposes the harmonic series.