# Series about inserting parentheses

Problem : Suppose that the series $$\sum_{}^{} a_k$$ converges and $${n_j}$$ is a strictly increasing sequence of positive integers. Define the sequence $$b_k$$ as follows :

$$b_1=a_1+...+a_{n_{1}}$$

$$b_2=a_{n_1+1}+...+a_{n_{2}}$$

...

$$b_k=a_{n_{k-1}+1}+...+a_{n_{k}}$$

Prove that $$\sum_{}^{} b_k$$ converges and that $$\sum_{}^{} a_k=$$ $$\sum_{}^{} b_k$$

My Proof :

Let $$s_m=\sum_{i=1}^{n_m} a_i$$ $$\forall m \in \mathbb{N}$$ then by definition, $$s_m=\sum_{i=1}^{n_m} a_i=\sum_{k=1}^{m} b_k$$

Since $$n_j$$ is strictly increasing sequence of positive integers, $$\lim_{m\to\infty}\ n_m= \infty$$

Since $$\lim_{m\to\infty}\ n_m= \infty$$ and $$\sum_{}^{} a_k$$ converges, $$\lim_{m\to\infty}\ s_m$$ exists and

$$\lim_{m\to\infty}\ s_m=$$ $$\lim_{m\to\infty}\sum_{i=1}^{n_m} a_i=$$ $$\sum_{}^{} a_k$$

Since $$\forall m \in \mathbb{N}$$ $$s_m=\sum_{k=1}^{m} b_k$$ and $$\lim_{m\to\infty}\ s_m$$ exists,

$$\sum_{}^{} b_k$$ converges and $$\sum_{}^{} a_k$$=$$\sum_{}^{} b_k$$

But, I'm not sure my proof is correct. Please give me feedback on my proof.

## 1 Answer

Yes, the idea of your proof is fine. However, your argument in the 2nd and 3rd line is unclear (among other things since $$n_j$$ is not defined). I suppose, in short your argument is:

Define $$s_n := \sum_{k = 0}^n a_k$$ and $$s_m := \sum_{k = 0}^m b_k$$. Then $$(s_m)_{m \in \mathbb{N}}$$ is a subsequence of $$(s_n)_{n \in \mathbb{N}}$$. Since $$\sum a_k$$ converges (by definition) $$(s_n)_{n \in \mathbb{N}}$$ converges, so does $$(s_m)_{m \in \mathbb{N}}$$ and thus $$\sum b_k$$ converges. Since limits are unique, they coincide.

Make those two lines precise and you are good to go.