The product of 2 infinite sums

[dyn]

Hi.
I know that e^ixe^-ix = 1 but if I write the product of the 2 exponentials as infinite series I get
Σ_nΣ_m 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]

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

 

[scottdave]

Hi.
I know that e^ixe^-ix = 1 but if I write the product of the 2 exponentials as infinite series I get
Σ_nΣ_m 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 e^u is Σ_n 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]

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

 

[Mark44]

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.