Is writing a sequence as a telescoping sum the only way to turn a sequence into an infinite series? In particular:
Let $s_n = 1 +\frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}- \ln(n+1), \ n \geq 1$. Convert this into an infinite series.
So let $a_n = s_n-s_{n-1}$. Then $a_n = \frac{1}{n}-\ln(n+1)+ \ln n$. This is equivalent to $$\sum_{n=1}^{\infty} \left(\frac{1}{n}- \ln \frac{n+1}{n} \right)$$
What does the inside term represent?