calculus help!?

Question

Prove that this series:

cosx = (1 - (x^2/2!) + (x^4/4!) - (x^6/6!) + ...) = summation notation symbol (with n=0 on bottom and infinity on top) of (-1)^n x^(2n)/(2n)!

represents cosx for all "x".

Answer 1

What is your definition of cos x? After all, the proof is that your definition matches the series.

I ask because the series is one of the standard definitions of cosine. Check out equation 4 in:
http://mathworld.wolfram.com/Cosine.html

If you can use:
e^(ix) = cos x + i sin x and
e^a = 1 + a + (1/2!)a^2 + (1/3!)a^3 + ...

Then substitute ix for a and you get real terms and imaginary terms (terms with a +i or -i). The sum of the real terms is cos x and is the same as the series.

If you can use calculus, then for any analytic function:

F(x) = F(0) + F'(0) x + (1/2!)F''(0)x^2 + ...
cos'(x) = - sin(x)
sin'(x) = cos(x)

so you can compute cos(x) in terms of powers of x and sin(0) and cos(0). The result will be the series you want.

Answer 2

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