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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
$$\limsup\limits_{n\to\infty} b_{n}\ \ \text{and}\ \ \liminf\limits_{n\to\infty} b_{n}$$

Please give some hint.

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    $\begingroup$ Compute $b_1,b_2,b_3,b_4$ and see a pattern. See the even and odd indices to make things clearer. $\endgroup$ – Teresa Lisbon Nov 12 '19 at 8:54
  • $\begingroup$ Not getting any pattern....b_{1}=1/2, b_{2}=1/2, b_{3}=1/6,.... $\endgroup$ – Sachin Nov 12 '19 at 8:57
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    $\begingroup$ 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? $\endgroup$ – Teresa Lisbon Nov 12 '19 at 9:00
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First study the 'inner workings' of the following two series:

$\tag 1 \displaystyle{\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}}$

$\tag 2 \displaystyle{\sum_{k=1}^{\infty} \frac{(-1)^{k}}{2}}$

The first series is known as the alternating harmonic series.

The second series doesn't converge, but you can still compute the $\text{lim sup}$ and $\text{lim inf}$
on the partial sums.

You'll also need to know some general identities that can be used when working with sigma summation.

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