telescoping sum

A telescoping sum is a sum in which cancellation occurs between subsequent terms, allowing the sum to be expressed using only the initial and final terms.

Formally a telescoping sum is or can be rewritten in the form

S=\\sum_{n=\\alpha}^{\\beta}\\left(a_{n}-a_{n+1}\\right)=a_{\\alpha}-a_{\\beta+1}

where $a_{n}$ is a sequence.

Example:

Define $S(N)=\\sum_{n=1}^{N}\\frac{1}{n(n+1)}$ . Note that by partial fractions of expressions:

\\frac{1}{n(n+1)}=\\frac{1}{n}-\\frac{1}{n+1}

and thus $a_{n}=\\frac{1}{n}$ in this example.

S(N)=\\sum_{n=1}^{N}\\left(\\frac{1}{n}-\\frac{1}{n+1}\\right)

=\\left(1-\\frac{1}{2}\\right)+\\cdots+\\left(\\frac{1}{n}-\\frac{1}{n+1}\\right)+%\\left(\\frac{1}{n+1}-\\frac{1}{n+2}\\right)+\\cdots+\\left(\\frac{1}{N}-\\frac{1}{N+1%}\\right)

=1+\\left(-\\frac{1}{2}+\\frac{1}{2}\\right)+\\cdots+\\left(-\\frac{1}{n+1}+\\frac{1}{%n+1}\\right)+\\cdots-\\frac{1}{N+1}

=1-\\frac{1}{N+1}