Telescopic Summation for Infinite Series

Telescopic Summation for Infinite Series. The finite telescoping idea of §5.5.2 extends immediately to infinite series: write $t_k=f(k)-f(k+1)$, note the finite-sum telescopes to $f(1)-f(n+1)$, and then let $n\to\infty$ — if $f(n+1)\to L$, the infinite sum is $f(1)-L$.

Example 5.20. For $\displaystyle\sum_{n=1}^\infty\frac1{n^2+5n+6}$: the partial-fraction split gives $a_n=\dfrac1{n^2+5n+6}=\dfrac1{n+2}-\dfrac1{n+3}$, so the partial sum telescopes:

$s_n = \left(\frac13-\frac14\right)+\left(\frac14-\frac15\right)+\cdots+\left(\frac1{n+2}-\frac1{n+3}\right) = \frac13-\frac1{n+3}.$ …