Closed-forms of infinite series with factorial in the denominator

Question

How to evaluate the closed-forms of series

\begin{equation} 1)\,\, \sum_{n=0}^\infty\frac{1}{(3n)!}\qquad\left|\qquad2)\,\, \sum_{n=0}^\infty\frac{1}{(3n+1)!}\qquad\right|\qquad3)\,\, \sum_{n=0}^\infty\frac{1}{(3n+2)!}\\ \end{equation}

Of course Wolfram Alpha can give us the closed-forms \begin{align} \sum_{n=0}^\infty\frac{1}{(3n)!}&=\frac{e}{3}+\frac{2\cos\left(\frac{\sqrt{3}}{2}\right)}{3\sqrt{e}}\\ \sum_{n=0}^\infty\frac{1}{(3n+1)!}&=\frac{e}{3}+\frac{2\sin\left(\frac{\sqrt{3}}{2}-\frac{\pi}{6}\right)}{3\sqrt{e}}\\ \sum_{n=0}^\infty\frac{1}{(3n+2)!}&=\frac{e}{3}-\frac{2\sin\left(\frac{\sqrt{3}}{2}+\frac{\pi}{6}\right)}{3\sqrt{e}} \end{align} but how to get those closed-forms by hand? I can only notice that \begin{equation} \sum_{n=0}^\infty\frac{1}{n!}=\sum_{n=0}^\infty\frac{1}{(3n)!}+\sum_{n=0}^\infty\frac{1}{(3n+1)!}+\sum_{n=0}^\infty\frac{1}{(3n+2)!}=e \end{equation} Could anyone here please help me? Any help would be greatly appreciated. Thank you.

PS: Please don't work backward.

Answer 1

Related techniques: (I). Here is an approach which enables you to tackle your problems. Let's consider the series

$$ f(x) = \sum_{n=0}^{\infty}\frac{x^{3n}}{(3n)!}. $$

Taking the Laplace transform gives

$$ F(s) = \sum_{n=0}^{\infty}\frac{1}{s^{3n+1}} = \frac{s^2}{s^3-1}. $$

To finish the problem you need to find the inverse Laplace of $F(s)$. One technique is partial fraction

$$ F(s) = \frac{1}{3(s-1)} + \frac{1}{3(s+1/2-i\sqrt{3}/2)} + \frac{1}{3(s+1/2+i\sqrt{3}/2)} .$$

Notes:

1) Laplace transform is defined as

$$ F(s) = \int_{0}^{\infty}f(x)e^{-sx}dx. $$

2) Laplace transform of $x^m$ is

$$ \frac{\Gamma(m+1)}{s^{m+1}} $$

3) Laplace transform of $e^{ax}$ is

$$ \frac{1}{s-a}. $$

Or equivalently the inverse Laplace of $\frac{1}{s-a}$ is $e^{ax}$

$$ $$

Answer 2

Another possible approach is to use the discrete Fourier transform. Let $\omega=\exp\frac{2\pi i}{3}$. Then: $$f(n)=\frac{1}{3}\left(1+\omega^n+\omega^{2n}\right)=\mathbb{1}_{n\equiv 0\!\pmod{3}}(n),$$ hence: $$\color{red}{\sum_{n=0}^{+\infty}\frac{1}{(3n)!}}=\sum_{n=0}^{+\infty}\frac{f(n)}{n!}=\frac{1}{3}\left(\exp(1)+\exp(\omega)+\exp(\omega^2)\right)=\color{red}{\frac{e}{3}+\frac{2}{3\sqrt{e}}\cos\frac{\sqrt{3}}{2}.}$$ The other two series can be computed with the same technique, by noticing that: $$f_1(n)=\frac{1}{3}\left(1+\omega^2\cdot\omega^n+\omega\cdot\omega^{2n}\right)=\mathbb{1}_{n\equiv 1\!\pmod{3}}(n),$$ $$f_2(n)=\frac{1}{3}\left(1+\omega\cdot\omega^n+\omega^2\cdot\omega^{2n}\right)=\mathbb{1}_{n\equiv 2\!\pmod{3}}(n).$$

Answer 3

$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ With $\ds{\ell = 0,1,2}$:

\begin{align} {\cal I}_{\ell}&\equiv\sum_{n = 0}^{\infty}{1 \over \pars{3n + \ell}!}= \sum_{n = 0}^{\infty}\sum_{k=0}^{\infty}{\delta_{k,3n + \ell} \over k!} =\sum_{n,k = 0}^{\infty}{1 \over k!} \oint_{\atop{\atop\verts{z}\ =\ a\ >\ 1}}{1 \over z^{3n + \ell - k + 1}} \,{\dd z \over 2\pi\ic} \\[3mm]&=\oint_{\atop{\atop\verts{z}\ =\ a\ >\ 1}}{1 \over z^{\ell + 1}} \bracks{\sum_{n = 0}^{\infty}\pars{1 \over z^{3}}^{n}} \bracks{\sum_{k = 0}^{\infty}{z^{k} \over k!}}\,{\dd z \over 2\pi\ic} \\[3mm]&=\oint_{\atop{\atop\verts{z}\ =\ a\ >\ 1}}{1 \over z^{\ell + 1}} \,{1 \over 1 - 1/z^{3}}\,\expo{z}\,{\dd z \over 2\pi\ic} =\oint_{\atop{\atop\verts{z}\ =\ a\ >\ 1}} {z^{2 - \ell} \over z^{3} - 1}\,\expo{z}\,{\dd z \over 2\pi\ic} \\[3mm]&=\sum_{m = -1}^{1}{z_{m}^{2 - \ell}\expo{z_{m}} \over 3z_{m}^{2}}\qquad \mbox{where}\qquad z_{m} \equiv \exp\pars{2m\pi\ic \over 3}\,,\quad m = -1,0,1 \end{align}

Then, \begin{align} {\cal I}_{\ell} &= {1 \over 3} \sum_{m = -1}^{1}z_{m}^{-\ell}\expo{z_{m}} ={1 \over 3}\,\expo{} + {2 \over 3}\,\Re\pars{z_{1}^{-\ell}\expo{z_{1}}} ={1 \over 3}\,\expo{} +{2 \over 3}\,\Re\pars{\expo{-2\ell\pi\ic/3}\exp\pars{\expo{2\pi\ic/3}}} \\[3mm]&={1 \over 3}\,\expo{} +{2 \over 3}\, \Re\pars{\expo{-2\ell\pi\ic/3}\exp\pars{-\,\half + {\root{3} \over 2}\,\ic}} \\[3mm]&={1 \over 3}\,\expo{} +{2 \over 3\root{\expo{}}}\, \Re\exp\pars{\bracks{{\root{3} \over 2} - {2\pi \over 3}\,\ell}\ic} \end{align}

$$ {\cal I}_{\ell}\equiv\sum_{n = 0}^{\infty}{1 \over \pars{3n + \ell}!} ={1 \over 3}\,\expo{} +{2 \over 3\root{\expo{}}}\,\cos\pars{{\root{3} \over 2} - {2\pi \over 3}\,\ell} \,,\qquad\ell = 0,1,2 $$

$$\begin{array}{rclcl} {\cal I}_{0}&=&\color{#66f}{\large\sum_{n = 0}^{\infty}{1 \over \pars{3n}!}} &=&{1 \over 3}\,\expo{} +{2 \over 3\root{\expo{}}}\,\cos\pars{{\root{3} \over 2}} \\[5mm] {\cal I}_{1}&=&\color{#66f}{\large\sum_{n = 0}^{\infty}{1 \over \pars{3n + 1}!}} &=&{1 \over 3}\,\expo{} +{2 \over 3\root{\expo{}}}\ \overbrace{\cos\pars{{\root{3} \over 2} - {2\pi \over 3}}} ^{\ds{\color{#c00000}{\sin\pars{{\root{3} \over 2} - {\pi \over 6}}}}} \\[5mm] {\cal I}_{2}&=&\color{#66f}{\large\sum_{n = 0}^{\infty}{1 \over \pars{3n + 2}!}} &=&{1 \over 3}\,\expo{} +{2 \over 3\root{\expo{}}}\ \underbrace{\cos\pars{{\root{3} \over 2} - {4\pi \over 3}}} _{\ds{\color{#c00000}{-\sin\pars{{\root{3} \over 2} + {\pi \over 6}}}}} \end{array} $$

Answer 4

This problem can be solved without advanced techniques. We have the Taylor series $$ e^x = \sum_{n\geq 0} \frac{x^n}{n!}. $$ As the questioner noted, plugging $x=1$ yields the equation $A+B+C=e$ connecting the three unknown sums. However, plugging in any cube root of unity also sheds light on the question because the numerators $x^n$ will repeat with period $3$. Let $\omega = e^{2\pi i/3}$; plugging $x = \omega$ yields $A + \omega B + \omega^2 C = e^{\omega}$, and plugging $x = \omega^2$ yields $A + \omega^2 B + \omega C = e^{\omega^2}$. We now have three equations that can be solved easily for the three unknowns $A$, $B$, and $C$. For instance, adding all three equations together yields $$ 3A = e + e^\omega + e^{\omega^2} = e + e^{-1/2}\left(\cos \frac{\sqrt{3}}{2} + i \sin \frac{\sqrt{3}}{2}\right) + e^{-1/2}\left(\cos \frac{-\sqrt{3}}{2} + i \sin \frac{-\sqrt{3}}{2}\right) $$ $$ = e + \frac{2}{\sqrt{e}} \cos \frac{\sqrt{3}}{2}, $$ in accordance with Wolfram Alpha.