Find $a_n$ using the partial sum of series

The sequence $(s_k)_{k \in \mathbb{N}}$ of partial sums of a series $\sum _{n=1}^{\infty}a_n$ is defined by $s_k= \sum _{n=1}^{k}a_n$. Consider a series $\sum _{n=1}^{\infty}a_n$ for which $s_k= \frac{k+1}{k} \ \forall k \in \mathbb{N}$. Find $a_n$.

What I have done is that I did $a_n=2$ for $n=1$, $a_n=\frac{k+1}{k}-\frac{k}{k-1}=-\frac{1}{k(k-1)}$ for all $n \ge 2$.

And I just want to know if there is a general expression that can define $a_n$ for all $n \in \mathbb{N}$.

Thanks to anybody who helps.

You need to fix your indices. If $k=1$ then $s_1 = a_1 = 2$. For $k \geq 2$ notice that

$$s_k - s_{k-1} = \sum_{n=1}^k a_n - \sum_{n=1}^{k-1} a_n = a_k$$

and

$$s_k - s_{k-1} = \frac{k+1}{k} - \frac{k}{k-1} = - \frac{1}{k(k-1)}.$$

Therefore

$$a_k = - \frac{1}{k(k-1)}.$$

• But then this only works for $k \ge 2$, if $k=1$ the fraction is undefined. – Lucy Nov 29 '14 at 17:24
• You already know the answer for $k=1$. All I did was correct the indices for the case $k \geq 2$. – Mark Fantini Nov 29 '14 at 17:26