Sum to infinity of An and Sk Consider a series $\sum_{n=1}^\infty A_n$ for which $S_k = \frac{k + 1}{k}$ $\forall k 
\in \mathbb N$. Find $A_n$ and the value of $\sum_{n=1}^\infty A_n$.
I think i have worked out that $\sum_{n=1}^\infty A_n$ is $1$. But i don't know how to work out $A_n$ or show fully how i got my answer for the sum to infinity. 
$S_k=\frac{k+1}{k}$
hence $\lim_{k\to \infty} S_k=1$. 
which implies that $\sum_{n=1}^\infty A_n$ is $1$.

By definition,
$$
\sum_{n=1}^\infty a_n = \lim_{n \to +\infty} \sum_{k=1}^n a_k.
$$
If you have really proved that
$$
S_n = \sum_{k=1}^n a_k = \frac{n+1}{n}
$$
for every $k$, then your conclusion is perfectly legitimate, since $\lim_{n \to +\infty} \frac{n+1}{n}=1$.
