On changing the order of summation between a finite sum and an infinite sum.

Question

Suppose that for every $j=1,...,t$ we have a convergent series of complex numbers $\displaystyle\sum_{m=0}^\infty a_j^m$. My question is:

Is it true in general that $$\sum_{j=1}^t\left(\sum_{m=0}^\infty a_j^m\right)=\sum_{m=0}^{\infty}\left(\sum_{j=1}^ta_j^m\right)$$ If not, under which extra conditions this can be true?

Thank you very much for your help.

Answer 1

It is always true because limits and finite sums commute.

$$\sum_{j=1}^t\sum_{m=0}^\infty a_j^m= \sum_{j=1}^t\lim_{n\to\infty} \sum_{m=0}^n a_j^m = \lim_{n\to\infty} \sum_{j=1}^t\sum_{m=0}^{n}a_j^m = \lim_{n\to\infty} \sum_{m=0}^{n} \sum_{j=1}^ta_j^m = \sum_{m=0}^{\infty}\sum_{j=1}^ta_j^m$$

Answer 2

It is not true in general for a technical reason, the expanded limit may not exist. For example, $\sum_n (2^n-2^n) = 0$ but $\sum_n 2^n -\sum_n 2^n$ doesn't make sense.

If $\sum_n a_n, \sum_n b_n$ are convergent, then we have $\sum_n (a_n+b_n) = \sum_n a_n + \sum_n b_n$ because the finite sums are interchangeable and one can then take limits.

Hence it is true for a finite collection of summations.