# $\limsup$ and $\liminf$ of a sequence

Consider a sequence $$a_{n}$$ with $$a_{n}=(-1)^{n} (\frac{1}{2}-\frac{1}{n})$$. Let $$b_{n}=\sum_{k=1}^{n} a_{k}$$ for all $$n\in\mathbb{N}$$. Then find
$$\limsup\limits_{n\to\infty} b_{n}\ \ \text{and}\ \ \liminf\limits_{n\to\infty} b_{n}$$

• Compute $b_1,b_2,b_3,b_4$ and see a pattern. See the even and odd indices to make things clearer. – Teresa Lisbon Nov 12 '19 at 8:54
• Not getting any pattern....b_{1}=1/2, b_{2}=1/2, b_{3}=1/6,.... – Sachin Nov 12 '19 at 8:57
• Note that $a_n$ is very close to half when $n$ is large even, and close to $-\frac 12$ when it is large and odd. Now, think about the sequences $b_{2n}$ and $b_{2n+1}$, can you see why they may be convergent? – Teresa Lisbon Nov 12 '19 at 9:00

$$\tag 1 \displaystyle{\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}}$$
$$\tag 2 \displaystyle{\sum_{k=1}^{\infty} \frac{(-1)^{k}}{2}}$$
The second series doesn't converge, but you can still compute the $$\text{lim sup}$$ and $$\text{lim inf}$$