Answer
$\ \sum_{k=1}^{13} \dfrac{k}{k+1}$
Work Step by Step
We need to compute summation formula for the sequence
$\dfrac{1}{2}+\dfrac{2}{3}+\dfrac{3}{4}+.......+\dfrac{13}{13+1}$
We see that there are $13$ positive terms as the numerator, take it as index $k$.
When $k=1$, the first term is $\dfrac{1}{2}$ and for $k=2$ , the second term is $\dfrac{2}{3}$.
These information indicate the $k^{th}$ term as $\dfrac{k}{k+1}$.
So, we write the the summation formula for kth term as follows:
$\dfrac{1}{2}+\dfrac{2}{3}+\dfrac{3}{4}+.......+\dfrac{13}{13+1}= \sum_{k=1}^{13} \dfrac{k}{k+1}$
