Variable substitution in sum's index

Question

How cam I simplify

Sum[UnitStep[-1 + n]/Sqrt[n!], {n, 0, ∞}]

to

$$\sum _{n=1}^{\infty } \frac{1}{\sqrt{n!}}$$

and then to

$$\sum _{n=0}^{\infty } \frac{1}{\sqrt{(n+1)!}}$$

I tried Simplify and FullSimplify, but neither of them worked.

Answer 1

Here is a brute force way, hope it will help.

changeindex = Sum[a_, b_] :> Module[{nstart, rule, body},
rule = 
 Reduce[And @@ 
   Thread[Flatten@Cases[a, UnitStep[x__] :> {x}, \[Infinity]] >= 
     0]];
nstart = Max[rule /. _ >= x_ -> x, 0];
body = 
 Evaluate[Simplify[a /. b[[1]] -> b[[1]] + nstart, b[[1]] >= 0]];
Sum[Evaluate@body, Evaluate@b]
];


In[1]:= Sum[UnitStep[-1 + n]/Sqrt[n!], {n, 0, \[Infinity]}] /. changeindex
Out[1]:=

$$\sum _{n=0}^{\infty } \frac{1}{\sqrt{(1+n)!}}$$