Easy Double Sums Question: $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{1}{(m+n)!}$

Question

How to calculate $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \dfrac{1}{(m+n)!} $ ?

I don't know how to approach it . Please help :)

P.S.I am new to Double Sums and am not able to find any good sources to study it , can anyone help please ?

Answer 1

This is the same as $$ \sum_{m=1}^\infty \sum_{k=m+1}^\infty \frac{1}{k!} $$ We can rearrange terms, noting that for each value of $k$ there will be terms only with $k > m$. There are $k-1$ possible values of $m$ that satisfy $k>m$. So $$ \sum_{m=1}^\infty \sum_{k=m+1}^\infty \frac{1}{k!} = \sum_{k=1}^\infty \frac{k-1}{k!} $$ The last trick is to note that it will be much easier to sum $\frac{k}{k!}$ so break up the numerator: $$ \sum_{k=1}^\infty \frac{k-1}{k!} = \sum_{k=1}^\infty \frac{k}{k!} - \sum_{k=1}^\infty \frac{1}{k!} = \sum_{k=1}^\infty \frac{1}{(k-1)!}- \sum_{k=1}^\infty \frac{1}{k!} $$ And this in turn is $$ \frac{1}{0!} + \sum_{k=2}^\infty \frac{1}{(k-1)!} - \sum_{k=1}^\infty \frac{1}{k!} = 1 +\sum_{j=1}^\infty \frac{1}{j!} - \sum_{k=1}^\infty \frac{1}{k!} $$ So far, only rearrangement of terms has happened. Now we note that $\sum_{k=1}^\infty \frac{1}{k!}$ is absolutely convergent, so the rearrangement of terms is valid; and the tow sums left cancel, so the answer is $$1 $$

Answer 2

Hint:

Under 'nice' conditions we have:

$$\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}a_{m+n}=\sum_{k=2}^{\infty}\left(k-1\right)a_{k}$$

Answer 3

$$\sum_{m,n=1}^{+\infty}\frac{1}{(m+n)!}=\sum_{h\geq 2}\frac{f(h)}{h!}$$ where: $$ f(h)=\#\left\{(m,n)\in\mathbb{N}_{\geq 1}^2:m+n=h\right\}=h-1 $$ hence: $$\sum_{m,n=1}^{+\infty}\frac{1}{(m+n)!}=\sum_{h\geq 2}\frac{h}{h!}-\sum_{h\geq 2}\frac{1}{h!}=\sum_{h\geq 1}\frac{1}{h!}-\sum_{h\geq 2}\frac{1}{h!}=\color{red}{1}.$$

Answer 4

\begin{align} &\sum_{m=1}^\infty \sum_{n=1}^\infty \frac1{(m+n)!}\\ =& \sum_{n=1}^\infty \frac1{(1+n)!} + \sum_{n=1}^\infty \frac1{(2+n)!} + \sum_{n=1}^\infty \frac1{(3+n)!} + \dots\\ =&\vphantom{+}\frac1{2!} + \frac1{3!} + \frac1{4!} + \frac1{5!} + \dots\\ &+\frac1{3!} + \frac1{4!} + \frac1{5!} + \frac1{6!} + \dots\\ &+\frac1{4!} + \frac1{5!} + \frac1{6!} + \frac1{7!} + \dots \end{align}

Let's add and subtract $\sum\limits_{j=1}^\infty \frac1{j!}$. It doesn't change the sum and it's easier to add.

\begin{align*} =&\left(\frac1{1!}+\color{red}{\frac1{2!}}+\color{green}{\frac1{3!}}+\color{blue}{\frac1{4!}}+\dots\right.\\ &+\color{red}{\frac1{2!}}+\color{green}{\frac1{3!}}+\color{blue}{\frac1{4!}}+\frac1{5!}+\dots\\ &+\color{green}{\frac1{3!}}+\color{blue}{\frac1{4!}}+\frac1{5!}+\frac1{6!}+\dots\\ &+\color{blue}{\frac1{4!}}+\frac1{5!}+\frac1{6!}+\frac1{7!}+\dots\\ &\left.\vphantom{\frac12}+\dots\right)\\ &-\frac1{1!}-\frac1{2!}-\frac1{3!}-\frac1{4!}-\dots\\ &=\left(1+\color{red}{\frac1{1!}}+\color{green}{\frac1{2!}}+\color{blue}{\frac1{3!}}+\dots\right)-\left(\frac1{1!}+\frac1{2!}+\frac1{3!}+\dots\right)=\boxed{1} \end{align*}

Hope this helps! I know that the adding and subtracting $\sum\frac1{j!}$ came out of nowhere, but it works.