Writing a sum in sigma notation

Each term has the form $\frac{i}{i+1}$, since the numerator increases $1, 2, 3, \dots$ while the denominator is always one more than the numerator: $2, 3, 4, \dots$. Checking against the given sum confirms this pattern works for every term.

The first term is $\frac{1}{2}$, which corresponds to $i=1$, and the last term is $\frac{9}{10}$, which corresponds to $i=9$. So the sum can be written as $\sum_{i=1}^{9} \frac{i}{i+1}.$