Interchanging the order of infinite and finite sum

Question

I have the following series $$f\left( {k,t} \right) = \sum\limits_{n = 0}^\infty {{a_n}(k){t^n}}.$$ It is obvious that $$\sum\limits_{k = 0}^m {f\left( {k,t} \right)} = \sum\limits_{k = 0}^m {\sum\limits_{n = 0}^\infty {{a_n}(k){t^n}} }.$$ But I don't know if the following is true or not $$\sum\limits_{k = 0}^m {\sum\limits_{n = 0}^\infty {{a_n}(k){t^n}} } = \sum\limits_{n = 0}^\infty {\sum\limits_{k = 0}^m {{a_n}(k){t^n}} }$$ I will appreciate if someone tell me under what conditions I can change the order of this double series.

Answer 1

That is not always true.

Take $t=1$, $m=1$, $a_n(0)= (-1)^n$ and $a_n(1)=(-1)^{n+1}$. The equality $$\sum\limits_{k = 0}^m {\sum\limits_{n = 0}^\infty {{a_n}(k){t^n}} } = \sum\limits_{n = 0}^\infty {\sum\limits_{k = 0}^m {{a_n}(k){t^n}} }$$ is not satisfied as $\sum (-1)^n$ doesn't converge.