Determine whether the series is convergent or divergent by expressing $s _ { n }$ as a telescoping sum. If it is convergent, find its sum. $\sum _ { n = 1 } ^ { \infty } \frac { 10 } { n ^ { 2 } + 2 n }$ .
A) $\frac { 15 } { 2 }$
B) diverges
C) $2$
D) $\frac { 1 } { 10 }$
E) $\frac { 2 } { 15 }$

Question

Quizplus · Accepted Answer

We can start by noticing that:
$$\frac {10} {n ^2 + 2n}=\frac {5} {n} - \frac {5} {n+2}$$
To see why, simply find a common denominator and combine the fractions on the right-hand side. Then, we can write:
$$\sum _ { n = 1 } ^ { \infty } \frac {10} {n ^2 + 2n} = \sum _ { n = 1 } ^ { \infty }\left (\frac {5} {n} - \frac {5} {n+2} \right )$$
At this point, we can group the terms in pairs and notice that most of them cancel out:
$$ \left (\frac {5} {1} - \frac {5} {3} \right ) +  \left (\frac {5} {2} - \frac {5} {4} \right ) + \left (\frac {5} {3} - \frac {5} {5} \right ) + \cdots$$
$$= 5 \left ( 1 - \frac {1} {2} + \frac {1} {2} - \frac {1} {3} + \frac {1} {3} - \frac {1} {4} + \cdots \right ) - \left ( \frac {5} {3} + \frac {5} {4} + \cdots \right )$$
All the cancellation makes this look like a telescoping sum. Indeed, we can simplify it further:
$$= 5(1) - \frac {5} {2}$$
$$= \frac {15} {2}$$
Thus, the series converges to $\frac {15} {2}$, which is option A.

Determine Whether the Series Is Convergent or Divergent by Expressing sns _ { n }sn​

Determine Whether the Series Is Convergent or Divergent by Expressing $s _ { n }$