How did this expansion take place?

[M. next]

Ok suppose we have e$^{x}$, by Taylor expansion it becomes $\sum$ x$^{n}$/n!
Then it was set equal to: $\sum$ x $^{2n}$/2n! + $\sum$ x $^{2n+1}$/(2n+1)!

Note: summation in all was from zero to infinity..
How did this take place??
Thanks

 

[cepheid]

All it's doing is splitting the series into its even and odd terms. Think about it: for n = 0, the first expression gives you x⁰/0! and the second one gives you x¹/1!

For n=1, the first expression gives you x²/2! And the second expression gives you x³/3!

Et cetera...

Between the two new series, all the terms from the original Taylor series are included.

 

[M. next]

Thank you a lot for this reply, but please bear with me and explain this sentence, " Between the two new series, all the terms from the original Taylor series are included."
How come?

 

[cepheid]

Because the first series has all the even terms from the original series, and the second one had all the odd terms from the original series.

I wrote out the first few terms for you already, hoping to illustrate this. TRY IT for yourself. Write out a few more terms.

 

[M. next]

Yes, you did. Thank you a lot :)!

 

[jbunniii]

By the way, beware that in general you cannot split an infinite series into its even and odd terms like that, without potentially changing the answer. It is only guaranteed to work if the series is absolutely convergent. Therefore some justification is needed before making manipulations like that. Fortunately, all power series (including Taylor series) are absolutely convergent within the interior of their radius of convergence. But convergence may only be conditional for points at the radius itself.

 

[M. next]

Thank you for the information. I will be careful when dealing with those.