Series - The Basics — Question 6

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

1. List the first six terms and partial sums.

2. Prove that the sequence of partial sums is bounded but divergent.

3. Apply the nth-term test and explain why bounded partial sums alone do not imply convergence.

*Question 6 – Solution

The series terms are $a_n=(-1)^n$, so the first six are $-1,1,-1,1,-1,1$. Adding successively gives the partial sums $$-1,0,-1,0,-1,0,\ldots.$$ More precisely, $$s_{2k}=0,\qquad s_{2k-1}=-1.$$

The sequence $(s_N)$ is bounded between $-1$ and $0$, but $s_{2k}=0\to0$ while $s_{2k-1}=-1\to-1$. Since these subsequences have different limits, $(s_N)$ does not converge. By definition, the series therefore diverges.

The terms $(-1)^n$ do not approach zero, so the nth-term test also proves divergence immediately. Indeed, if $s_N\to S$, then $a_N=s_N-s_{N-1}\to S-S=0$. This example shows that bounded partial sums are not enough; the partial sums must approach one number.