Summation of series- substitution If we have $\sum_{n=0}^{\infty}nf(n)=C, C\ne0\tag 1$, C is a constant,
can we find a closed form for f(n)?.
NB :
 Given condition is that $\sum_{n=0}^{\infty}f(n)$ converges to a constant value $K$ and $f(n)$ tends to $0$ when n goes to infinity. $f(0)=3,f(1)=5$ Thanks

Let $f(n)=0$ for $n>4$. Then what we have to solve is:
$$\left\{\begin{matrix}
(0\cdot f(0)+1\cdot f(1))+2f(2)+3f(3)+4f(4)&=&C \\
(f(0)+f(1))+f(2)+f(3)+f(4)&=&K
\end{matrix}\right.$$
These simultaneous equations have infinitely many solutions $(f(2),f(3),f(4))$.
What I want to say is, can $f(n)$ be determined uniquely?
