The Indefinite(f, k) command computes a closed form of the indefinite sum of $f$ with respect to $k$.

The command uses a combination of algorithms handling various classes of summands. They include the classes of polynomials, rational functions, and hypergeometric terms and j-fold hypergeometric terms, functions for which their minimal annihilators can be constructed, for example, d'Alembertian terms. For more information, see LinearOperators.

A library extension mechanism is also used to include sums for which an algorithmic approach to finding closed forms does not yet exist. Currently the computable summands include expressions containing the harmonic function, logarithmic function, digamma and polygamma functions, and sin, cos, and the exponential functions.

If the option failpoints=true (or just failpoints for short) is specified, then the command returns a pair $s,\left[p\,q\right]$, where

$s$ is a closed form of the indefinite sum of $f$ w.r.t. $k$, as above,

$p$ is a list of intervals $\left[a_1..b_1\,a_2..b_2\,\mathrm{...}\,a_m..b_m\right]$ where $f$ does not exist, and

$q$ is a list of points $k_i$ where the computed sum $s$ does not satisfy the telescoping equation $s\left(k_i+1\right)-s\left(k_i\right)=f\left(k_i\right)$ or does not exist.

If such points appear in the summation interval, the discrete Newton-Leibniz formula may fail.

If the command is unable to compute one of the lists $p,q$, it returns $s,\mathrm{FAIL}$.

$\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{IndefiniteSum}\right]\right)\:$

$f≔\frac{1}{n^2+\mathrm{sqrt}\left(5\right)n-1}$

$f≔\frac{1}{n^2+\sqrt{5}n-1}$

$s≔\mathrm{Indefinite}\left(f\,n\right)$

$s≔-\frac{1}{3\left(n-\frac{3}{2}+\frac{\sqrt{5}}{2}\right)}-\frac{1}{3\left(n-\frac{1}{2}+\frac{\sqrt{5}}{2}\right)}-\frac{1}{3\left(n+\frac{1}{2}+\frac{\sqrt{5}}{2}\right)}$

$\mathrm{evala}\left(\mathrm{Normal}\left(\mathrm{eval}\left(s\,n=n+1\right)-s\right)\,\mathrm{expanded}\right)$

$\frac{1}{n^2+\sqrt{5}n-1}$

$f≔\frac{2^{2n-1}}{n\left(2n+1\right)\mathrm{binomial}\left(2n\,n\right)}$

$f≔\frac{2^{2n-1}}{n\left(2n+1\right)\left(\genfrac{}{}{0ex}{}{2n}{n}\right)}$

$s≔\mathrm{Indefinite}\left(f\,n\right)$

$s≔-\frac{2^{2n-1}}{\left(\genfrac{}{}{0ex}{}{2n}{n}\right)n}$

$\mathrm{normal}\left(\mathrm{expand}\left(\mathrm{eval}\left(s\,n=n+1\right)-s\right)\right)$

$\frac{{\left(2^n\right)}^2}{2n\left(2n+1\right)\left(\genfrac{}{}{0ex}{}{2n}{n}\right)}$

$f≔\frac{1}{5}{\left({\left(\frac{1}{2}+\frac{1}{2}{5}^{\frac{1}{2}}\right)}^n-{\left(\frac{1}{2}-\frac{1}{2}{5}^{\frac{1}{2}}\right)}^n\right)}^2$

$f≔\frac{{\left({\left(\frac{1}{2}+\frac{\sqrt{5}}{2}\right)}^n-{\left(\frac{1}{2}-\frac{\sqrt{5}}{2}\right)}^n\right)}^2}{5}$

$\mathrm{Indefinite}\left(f\,n\right)$

$-\frac{3{\left({\left(\frac{1}{2}+\frac{\sqrt{5}}{2}\right)}^n-{\left(\frac{1}{2}-\frac{\sqrt{5}}{2}\right)}^n\right)}^2}{10}-\frac{{\left({\left(\frac{1}{2}+\frac{\sqrt{5}}{2}\right)}^{n+1}-{\left(\frac{1}{2}-\frac{\sqrt{5}}{2}\right)}^{n+1}\right)}^2}{10}+\frac{{\left({\left(\frac{1}{2}+\frac{\sqrt{5}}{2}\right)}^{n+2}-{\left(\frac{1}{2}-\frac{\sqrt{5}}{2}\right)}^{n+2}\right)}^2}{10}$

$\mathrm{sum}\left(\ln \left(2n+1\right)\,n\right)\mathbf{assuming}0\le n$

$n\ln \left(2\right)+\ln \left(\mathrm{\Gamma }\left(n+\frac{1}{2}\right)\right)$

$\mathrm{sum}\left(\ln \left(2n+1\right)\,n\right)\mathbf{assuming}n\le -1$

$n\ln \left(2\right)-\ln \left(\mathrm{\Gamma }\left(\frac{1}{2}-n\right)\right)+\mathrm{I}\mathrm{\pi }n$

$f≔\sin \left(n\right)\cos \left(n+1\right)-\mathrm{\Psi }\left(n\right)$

$f≔\sin \left(n\right)\cos \left(n+1\right)-\mathrm{\Psi }\left(n\right)$

$\mathrm{Indefinite}\left(f\,n\right)$

$-\frac{\sin \left(1\right)n}{2}-\frac{\csc \left(1\right){\cos \left(n\right)}^2}{2}-\mathrm{\Psi }\left(n+1\right)n+n+\mathrm{\Psi }\left(n\right)+\mathrm{\gamma }$

$f≔nn!$

$f≔nn!$

$\mathrm{Indefinite}\left(f\,n\,'\mathrm{failpoints}'\right)$

$n!,\left[\left[-\mathrm{\infty }..\mathrm{-1}\right]\,\left[\right]\right]$

$\mathrm{eval}\left(f\,n=-3\right)$

Error, numeric exception: division by zero

$\mathrm{limit}\left(f\,n=-3\right)$

$\mathrm{undefined}$

$f≔\frac{1}{n}+\frac{1}{n-1}-\frac{2}{n-5}$

$f≔\frac{1}{n}+\frac{1}{n-1}-\frac{2}{n-5}$

$s,\mathrm{fp}≔\mathrm{Indefinite}\left(f\,n\,'\mathrm{failpoints}'\right)$

$s,\mathrm{fp}≔\frac{2}{n-5}+\frac{2}{n-4}+\frac{2}{n-3}+\frac{2}{n-2}+\frac{1}{n-1},\left[\left[0..1\,5..5\right]\,\left[2\,3\,4\right]\right]$

$\mathrm{limit}\left(s\,n=2\right)$

$\mathrm{undefined}$

$f≔\frac{\mathrm{binomial}\left(2n-3\,n\right)}{4^n}$

$f≔\frac{\left(\genfrac{}{}{0ex}{}{2n-3}{n}\right)}{4^n}$

$s,\mathrm{fp}≔\mathrm{Indefinite}\left(f\,n\,'\mathrm{failpoints}'\right)$

$s,\mathrm{fp}≔\frac{2n\left(n+1\right)\left(\genfrac{}{}{0ex}{}{2n-3}{n}\right)}{\left(n-2\right)4^n},\left[\left[\right]\,\left[2\right]\right]$

$\mathrm{eval}\left(s\,n=2\right)$

Error, numeric exception: division by zero

$\mathrm{limit}\left(s\,n=2\right)$

$\frac{3}{8}$

$\left(\mathrm{limit}\left(s\,n=2\right)\right)-\mathrm{eval}\left(s\,n=1\right)=\mathrm{eval}\left(f\,n=1\right)$

$-\frac{5}{8}=-\frac{1}{4}$

$\mathrm{sum}\left(f\,n=0..10\right)=\mathrm{eval}\left(s\,n=11\right)-\mathrm{eval}\left(s\,n=0\right)$

$\frac{236871}{262144}=\frac{138567}{262144}$

$\mathrm{f1}≔\mathrm{convert}\left(f\,\mathrm{\Gamma }\right)$

$\mathrm{f1}≔\frac{\mathrm{\Gamma }\left(2n-2\right)}{\mathrm{\Gamma }\left(n+1\right)\mathrm{\Gamma }\left(n-2\right)4^n}$

$s,\mathrm{fp}≔\mathrm{Indefinite}\left(\mathrm{f1}\,n\,'\mathrm{failpoints}'\right)$

$s,\mathrm{fp}≔\frac{2n\left(n+1\right)\mathrm{\Gamma }\left(2n-2\right)}{\left(n-2\right)\mathrm{\Gamma }\left(n+1\right)\mathrm{\Gamma }\left(n-2\right)4^n},\left[\left[\right]\,\left[\right]\right]$

$\mathrm{sum}\left(\mathrm{f1}\,n=0..10\right)=\mathrm{eval}\left(s\,n=11\right)-\left(\mathrm{limit}\left(s\,n=0\right)\right)$

$\frac{138567}{262144}=\frac{138567}{262144}$

$\mathrm{\_EnvFormal}≔\mathrm{false}\:$

$s,\mathrm{fp}≔\mathrm{Indefinite}\left(\mathrm{f1}\,n\,'\mathrm{failpoints}'\right)$

$s,\mathrm{fp}≔\frac{2n\left(n+1\right)\mathrm{\Gamma }\left(2n-2\right)}{\left(n-2\right)\mathrm{\Gamma }\left(n+1\right)\mathrm{\Gamma }\left(n-2\right)4^n},\left[\left[-\mathrm{\infty }..2\right]\,\left[\right]\right]$

$\mathrm{sum}\left(\mathrm{f1}\,n=0..10\right)$

Error, (in SumTools:-DefiniteSum:-ClosedForm) summand is singular in the interval of summation

$\mathrm{\_EnvFormal}≔'\mathrm{\_EnvFormal}'\:$