# I The product of 2 infinite sums

#### dyn

Hi.
I know that eixe-ix = 1 but if I write the product of the 2 exponentials as infinite series I get
ΣnΣm xn/(n!) (-x)m/(m!)
without knowing the result is 1 using exponentials how would I get the result of this product of 2 infinite sums ?
Thanks

#### fresh_42

Mentor
2018 Award
What do you get, if you do not ignore the $i$ as you did, and group the products by degree of $x$?

#### scottdave

Homework Helper
Gold Member
Hi.
I know that eixe-ix = 1 but if I write the product of the 2 exponentials as infinite series I get
ΣnΣm xn/(n!) (-x)m/(m!)
without knowing the result is 1 using exponentials how would I get the result of this product of 2 infinite sums ?
Thanks
So the series for eu is Σn un/(n!)
and u is either ix or -ix (in your example). But you lost the i when converting to the series.

#### dyn

Yes sorry I forgot the i when writing out the infinite series

#### WWGD

Gold Member
You may also use: $e^{ix}=(cosx+isinx)$ and symmetry properties of $sinx, cosx$.

#### Mark44

Mentor
You may also use: $e^{ix}=(cosx+isinx)$ and symmetry properties of $sinx, cosx$.
Sure, you could do that for this particular problem, but this ignores the OP's question about the product of two infinite series.

"The product of 2 infinite sums"

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