How to solve double summations?

Hi everyone!

Can anyone tell me how to solve problems of double summation?

What are tricks to solve them? For instance how to solve this problem:

\[\sum_{ω=1}^{\infty}\sum_{n=1}^{\infty}\frac{ω^{2}n}{3^{n}(n(3^{ω})+ω(3^{n}))}\] Also help with generalised methods .

#Calculus

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Not every double sum has a closed form, just as not every single summation has a closed form. Many summation solutions are ad hoc.

Jake Lai - 5 years, 3 months ago

What is ad hoc? Can you explain me general properties to solve double summation using an example?

Shivam Jadhav - 5 years, 3 months ago

"Ad hoc" is a term used to describe methods/solutions that are non-generalizable and applicable to a particular case.

Prasun Biswas - 5 years, 3 months ago

The answer is $\displaystyle\frac{9}{32}$ . First check the absolute convergence of the repeated series. Then you can interchange the order of summation at some stage. The repeated series turns out to be $\displaystyle\frac{1}{2}\left(\displaystyle\sum_{n=1}^{\infty}\frac{n}{3^n}\right)^2$ , which is easy to deal with.

Haosen Chen - 11 months, 2 weeks ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$