Semantic question about summations

Question

This is a simple question about the syntax of summations. Is the following true:

$\lambda_1\sum_\limits{j=1}^{\infty} f_1(E_j)+ \lambda_2\sum_\limits{j=1}^{\infty}f_2(E_j)...+\lambda_n\sum_\limits{j=1}^{\infty}f_n(E_j)=\sum_\limits{j=1}^{\infty}\sum_\limits{i=1}^{n}\lambda_if_i(E_j)$

@Chappers Even though I'd be moving a scalar into an infinite sum? — Jimmy2Goons, Apr 28 '15 at 02:12
The problem you're going to run into there is if only one series on the left diverges and its multiplier is zero, or if multiple of the series on the left diverge. Then the left hand side is unlikely to make sense. The right-hand side always makes sense, although of course it may diverge. — Chappers, Apr 28 '15 at 02:24

score 2 · Answer 1 · answered Apr 28 '15 at 02:52

This is not a "semantic" issue, this is a real analysis issue.

If every series on the left converges, then the series on the right converges, and has the same sign. That's about all you can say.

To analyze this carefully, let $S_k(m) = \lambda_k \sum_{j=1}^m f_k(E_j)$. Your question is equivalent to $$\sum_{k=1}^n \lim_{m\to\infty} S_k(m) \overset{?}{=} \lim_{m\to\infty} \sum_{k=1}^n S_k(m)$$ A standard theorem says that the sum of convergent sequences converges.

On the other hand, it's possible for the expression on the right to converge without anything on the left making sense. For example (with $n=2$) let $S_1(m)=m$, $S_2(m)=-m$.

Haha. Good call. Ran into this on a measure theory assignment. Felt like an idiot for not knowing the answer. — Jimmy2Goons, Apr 28 '15 at 03:23

Semantic question about summations

1 Answers1