# How can I get the simplest result of this sum?

I am trying to find the sum $$\sum _{k=1}^n \frac{k+1}{(k-1)!+k!+(k+1)! }.$$ I tried

Simplify[Sum[(k + 1)/(( k - 1)! + k! + (k + 1)!), {k, 1, n}]]


and got

(-4 - 4 n - n^2 + (1 + n) n! + (1 + n) (1 + n)! + (2 + n)! + n (2 + n)!)/((1 + n) (n! + (1 + n)! + (2 + n)!))

and tried

FullSimplify[Sum[(k + 1)/(( k - 1)! + k! + (k + 1)!), {k, 1, n}]]


1 - 1/Gamma[2 + n]

How can I get the result like Maple?

• Note that with the default complexity function, definition given as SimplifyCount in the documentation on ComplexityFunction, 1 - 1/Gamma[2 + n] has the same complexity as 1 - 1/Factorial[1 + n] (12). Commented Aug 23, 2018 at 0:58

Simplify@SimplifySimplifyGamma[Sum[(k + 1)/((k - 1)! + k! + (k + 1)!), {k, 1, n}]]
% // TeXForm


$1-\frac{1}{(n+1)!}$

or

Simplify @ DeveloperGammaSimplify[Sum[(k + 1)/((k - 1)! + k! + (k + 1)!), {k, 1, n}]]
% // TeXForm


$1-\frac{1}{(n+1)!}$

Mathematica's result

1 - 1/Gamma[2 + n]


is equivalent to Maple's for $n \in \mathbb{Z}$, but is also a correct generalization for non-integer $n$.