Is this convergent sum a constant?

Question

I got this symbolic convergent sum from $\textit{Mathematica}$: $$\sum _{k=1}^{\infty } \frac{k!}{(2 k)!}=\frac{1}{2} \sqrt[4]{e} \sqrt{\pi } \text{erf}\left(\frac{1}{2}\right)$$ Where $\text{erf}\left(\frac{1}{2}\right)$ can be found here.

Is this convergent sum a constant? I'm guessing "yes," but I have never encountered this kind of sum before.

What do you mean that it is a constant? The sum does not depend on anything, hence it is a number... — Dennis Gulko, Mar 10 '13 at 10:20
@DennisGulko, exactly what I thought. Make this an answer and I'll sign off on it. — Fred Daniel Kline, Mar 10 '13 at 10:23

score 2 · Answer 1 · answered Mar 10 '13 at 10:23

The LHS of the equality is a convergent series, therefore it is real number (or a constant). This means that the RHS has got to be a constant. Since $\displaystyle \frac{1}{2} \sqrt[4]{e} \sqrt{\pi }$ is a constant, then $\operatorname{erf} \left(\frac{1}{2}\right)$ must also be a constant.

Another way to look at it is (if you know the error function), it's just realising that $\displaystyle \operatorname{erf}\left(\frac{1}{2}\right)$ is simply the error function $\operatorname{erf}(x)$ evaluated at $\displaystyle x=\frac{1}{2}$ (which is part of its domain), hence a constant.

score 2 · Accepted Answer · answered Mar 10 '13 at 10:25

2

Following the request by the OP, I'm posting this as an answer:

 The sum does not depend on anything, hence it is a number

answered Mar 10 '13 at 10:25

Dennis Gulko

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Is this convergent sum a constant?

2 Answers2