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In my calculus notes, my professor presented the following proposition:

Proposition: Let $\sum_{n=1}^{\infty}a_n$ be a convergent series and let $1=n_1<n_2<n_3<...$ be an increasing sequence of indices. Consider the set of grouped sums of the sequence between these indices, i.e; $$A_k=\sum_{n=n_k}^{n_{k+1}-1}a_n.$$ Then, $$\sum_{k=1}^{\infty}A_k=\sum_{n=1}^{\infty}a_n$$

This proposition is supposed to say that if an infinite series converges, then the associative property holds. I was reading an equivalent proposition, but written in a different way, and I was able to prove it. However, I can't prove the statement the way my professor wrote it. Any help is welcome

Addendum: The proposition I was able to prove is Exercise 2.5.3 (a) on page 65 from the book Understanding Analysis, 2nd ed by Stephen Abbott.

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3 Answers 3

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Hint: $$\sum_{k=1}^N A_k =\sum_{i=1}^{n_{N+1}-1} a_i$$ because finite sums are associative.

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It's easy to see that the sequence of partial sums of the $A_k$ is Cauchy, so that sequence of partial sums (i.e., the infinite sum) must converge to something. And since the sequence of partial sums of the $A_k$ is a subsequence of the sequence of partial sums of $a_n$, we know what that something has to be.

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May be following view also will be helpful: $\sum\limits_{k=1}^N A_k$ is subsequence of $\sum\limits_{k=1}^N a_k$.

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