# A question about Taylor Series

Find the Taylor series for cosx and indicate why it converges to cosx for all x in R.

The Taylor series for cosx can be found by differentiating $sum_{k=0}^{\infty} \frac{(-1)^k (x^{2k+1})}{(2k+1)!}$ on both sides...

But I'm not sure what the question means by "why it converges to cosx for all x in R". Isn't that just obvious, since the summation equals cosx...it obviously converges to it...

tiny-tim
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Hi Artusartos! (you could try comparing it with the definition of eix)

Hi Artusartos! (you could try comparing it with the definition of eix)

Thanks. I'm not sure how I can compare it to e^(ix), but this is how I did it:

I just found $$lim |\frac{a_{n+1}}{a_n}|$$ and got infinity as the radius of convergence...

tiny-tim
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yes, that should do! (and look carefully, and you'll se that the cosx series is the real part of ∑ (ix)n/n! )

HallsofIvy
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Be careful here! Showing that the radius of convergence is infinity shows that it converges for all x. That alone does not show that it converges to cos(x).

Be careful here! Showing that the radius of convergence is infinity shows that it converges for all x. That alone does not show that it converges to cos(x).

Thanks, but then what can I do to show that it converges to cos(x)?

tiny-tim
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Thanks, but then what can I do to show that it converges to cos(x)?

Euler's formula? (and look carefully, and you'll se that the cosx series is the real part of ∑ (ix)n/n! )

Euler's formula? Thanks but I'm not sure what you mean. Doesn't the fact that cosx is equal to the Taylor series imply that they all converge to cosx? tiny-tim
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Doesn't the fact that cosx is equal to the Taylor series imply that they all converge to cosx? if cosx is analytic, then yes

from http://en.wikipedia.org/wiki/Analytic_function
A function is analytic if and only if it is equal to its Taylor series in some neighborhood of every point.​

but that's a rather circular answer …

there must be theorems that state when a function is analytic, but i can't offhand remember what they are if cosx is analytic, then yes

from http://en.wikipedia.org/wiki/Analytic_function
A function is analytic if and only if it is equal to its Taylor series in some neighborhood of every point.​

but that's a rather circular answer …

there must be theorems that state when a function is analytic, but i can't offhand remember what they are Thanks, but we didn't learn about analytic functions yet...

Ray Vickson
Thanks but I'm not sure what you mean. Doesn't the fact that cosx is equal to the Taylor series imply that they all converge to cosx? $$f(x) = T_n(x) + E_n(x),$$