Telescoping - Maple Help

SumTools[DefiniteSum]

 Telescoping
 compute closed forms of definite sums using telescoping method

 Calling Sequence Telescoping(f, k=m..n)

Parameters

 f - expression; specified summand k - name; summation index m, n - expressions or integers

Description

 • The Telescoping(f, k=m..n) command computes a closed form of the definite sum of f over the specified range of k using telescoping method, or Newton-Leibniz's formula, that is, it first computes a closed form of the corresponding indefinite sum.

Examples

 > $\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{DefiniteSum}\right]\right):$
 > $F≔\mathrm{binomial}\left(2n-2k,n-k\right){2}^{4k}{\left(2k\left(2k+1\right)\mathrm{binomial}\left(2k,k\right)\right)}^{-1}$
 ${F}{≔}\frac{\left(\genfrac{}{}{0}{}{{2}{}{n}{-}{2}{}{k}}{{n}{-}{k}}\right){}{{2}}^{{4}{}{k}}}{{2}{}{k}{}\left({2}{}{k}{+}{1}\right){}\left(\genfrac{}{}{0}{}{{2}{}{k}}{{k}}\right)}$ (1)
 > $\mathrm{Sum}\left(F,k=1..n\right)=\mathrm{Telescoping}\left(F,k=1..n\right)$
 ${\sum }_{{k}{=}{1}}^{{n}}{}\frac{\left(\genfrac{}{}{0}{}{{2}{}{n}{-}{2}{}{k}}{{n}{-}{k}}\right){}{{2}}^{{4}{}{k}}}{{2}{}{k}{}\left({2}{}{k}{+}{1}\right){}\left(\genfrac{}{}{0}{}{{2}{}{k}}{{k}}\right)}{=}{-}\frac{\left(\genfrac{}{}{0}{}{{2}{}{n}{-}{2}}{{n}{-}{1}}\right){}\left({-}{16}{}{n}{+}{8}\right)}{{2}{}\left({2}{}{n}{+}{1}\right)}$ (2)