Question

Determine whether the series is convergent or divergent by expressing sn as a telescoping sum (as in example 7). ∞ cos 7 n2 − cos 7 (n + 1)2 n = 1

Answer 1

Looks like the series is

$\displaystyle\sum_{n=1}^\infty\cos\frac7{n^2}-\cos\frac7{(n+1)^2}$

Consider the $k$-th partial sum of the series:

$\displaystyle\sum_{n=1}^k\cos\frac7{n^2}-\cos\frac7{(n+1)^2}=$$\left(\cos7-\cos\dfrac7{2^2}\right)+\left(\cos\dfrac7{2^2}-\cos\dfrac7{3^3}\right)+\cdots+\left(\cos\dfrac7{k^2}-\cos\dfrac7{(k+1)^2}\right)$
We can see some cancellation occurring that leaves us with
$\displaystyle\sum_{n=1}^k\cos\frac7{n^2}-\cos\frac7{(n+1)^2}=\cos7-\cos\frac7{(k+1)^2}$
Then as $k\to\infty$, we have $\dfrac7{(k+1)^2}\to0$ and $\cos0=1$, so the series converges to $\cos7-1$.