91Ó°ÊÓ

Evaluate the following telescoping series or state whether the series diverges. \(\sum_{n=1}^{\infty} 2^{1 / n}-2^{1 /(n+1)}\)

Evaluate the following telescoping series or state whether the series diverges. \(\sum_{n=1}^{\infty}(\sqrt{n}-\sqrt{n+1})\)

Evaluate the following telescoping series or state whether the series diverges. \(\sum_{n=1}^{\infty}(\sin n-\sin (n+1))\)

Express the following series as a telescoping sum and evaluate its \(n\)th partial sum. \(\sum_{n=1}^{\infty} \ln \left(\frac{n}{n+1}\right)\)

Express the following series as a telescoping sum and evaluate its \(n\)th partial sum. \(\sum_{n=1}^{\infty} \frac{2 n+1}{\left(n^{2}+n\right)^{2}}\)

Express the following series as a telescoping sum and evaluate its \(n\)th partial sum. \(\sum_{n=1}^{\infty} \frac{(n+2)}{n(n+1) 2^{n+1}}\)

A general telescoping series is one in which all but the first few terms cancel out after summing a given number of successive terms. Let \(a_{n}=f(n)-2 f(n+1)+f(n+2),\) in which \(f(n) \rightarrow 0\) as \(n \rightarrow \infty .\) Find \(\sum_{n=1}^{\infty} a_{n}\).

Suppose that \(a_{n}=c_{0} f(n)+c_{1} f(n+1)+c_{2} f(n+2)+c_{3} f(n+3)+c_{4} f(n+4)\) where \(f(n) \rightarrow 0\) as \(n \rightarrow \infty\). Find a condition on the coefficients \(c_{0}, \ldots, c_{4}\) that make this a general telescoping series.

Evaluate \(\sum_{n=1}^{\infty} \frac{1}{n(n+1)(n+2)}\).

Evaluate \(\sum_{n=2}^{\infty} \frac{2}{n^{3}-n}\).