Taylor series! Help plz!!?

Question

HELP PLEASE...THANKS~~~
Find the taylor series centred at c = 0 for the function f(x) = ((e^x)+e^(-x))/2.

Answer 1

e^x = ∑ x^n / n!

e^(-x) = ∑ (-1)^n x^n / n!

e^x + e^(-x) = ∑ 2*x^(2n) / (2n)!
[e^x + e^(-x)]/2 = ∑ x^(2n) / (2n)!

Answer 2

Of course, Dr. D's answer is absolutely correct (and elegant). But it occurs to me that you may not know the MacLaurin series for e^x (and hence for e^(-x)). On the other hand, if you did know the series for e^x, you would also know the series for sinh x, which is the given function.

If in fact you do not know those basic series, then you would have to compute the series yourself. The general term in a Taylor series for f(x) at c=0 is (f(0))^(n)*x^n/n!, where (f(0))^(n)
denotes the n-th derivative evaluated at 0.

You will easily find the pattern. The n-th derivative is 1 when n is even, and 0 when n is odd.