Hi, how can one find the sum of the series:
for the sequence such that:
Is it some series?
Thanks.
I don't fully understand your question, but maybe this would be helpful.
$\displaystyle S_n(x)=\sum^n_{k=1}a_kx^k$
$\displaystyle S_n$ is clearly $\displaystyle S_n(1)$.
Now, the derivative of $\displaystyle S_n(x)$ is:
$\displaystyle (S_n(x))'=(\sum^n_{k=1}a_kx^k)'=\sum^n_{k=1}ka_kx^ {k-1}$.
So $\displaystyle (S_n(1))'=\sum^n_{k=1}ka_k $.