Hi everyone. I'm doing a few tests with infinite sums in mathematica, but I have troubles. I want to manipulate formal power series with non fixed coefficient.

For exemple, defining $f(t)=\sum_{n=0}^\infty u_n t^n$ and $g(t)=\sum_{n=0}^\infty v_n t^n$. I would like Mathematica to regroup the sums when computing $f(t)+g(t)$, i.e. I would like Mathematica to give me the output :

$$\sum_{n=0}^\infty ( u_n + v_n ) t^n$$

I tried the following code :

f[t_]:=Sum[Subscript[u,n]*t^n,{n,1,Infinity}];
g[t_]:=Sum[Subscript[v,n]*t^n,{n,1,Infinity}];
f[t]+g[t]

But he give me the result as the sum of two separated infinite sums, without regrouping them. I also tried Simplify or FullSimplify, but it didn't give what I wanted. What surprizes me is that for example D[f[t],t] give the formal output I would expect : $$\sum_{n=0}^\infty n u_n t^{n-1}$$

In the end, I would also like Mathematica to be able to change the index and then regrouping the sums. Typically, I'm looking for things like : $$f(t)+f'(t)=\sum_{n=0}^\infty (u_n+(n+1)u_{n+1})t^n$$

Would this be possible ? Thank you.