Alternating Series Test — Question 6

*Question 6 Consider $\displaystyle\sum_{n=2}^{\infty}\frac{(-1)^n}{n+(-1)^n}$.

1. Explain the corrected starting index and why the usual AST magnitude is not decreasing.

2. Pair consecutive even and odd terms to test convergence.

3. Test absolute convergence and classify the series.

*Question 6 – Solution

At $n=1$ the denominator is $1-1=0$, so the series must start at $2$. Magnitudes satisfy $b_{2k}=1/(2k+1)$ and $b_{2k+1}=1/(2k)$; hence $b_{2k+1}>b_{2k}$ infinitely often, so the standard monotonic AST does not apply.

$$a_{2k}+a_{2k+1}=\frac1{2k+1}-\frac1{2k}=-\frac1{2k(2k+1)}.$$ The paired series converges absolutely by comparison with $1/(4k^2)$. Since the unpaired final term tends to zero, both even and odd partial-sum subsequences approach the same limit; the original series converges.

$$|a_{2k}|+|a_{2k+1}|=\frac1{2k+1}+\frac1{2k}\ge\frac1{2k}.$$ Thus the absolute series diverges, so convergence is conditional.