Question

Prove that sum of analytic function is again analytc

Answer 1

Explanation:

I know the fact that not every C∞ function is analytic, for which there is the famous example:

f(x):={ e−1/x x>0 0 x≤0

In that case, f is C∞ but its Taylor series is identically zero, which is clearly different from f itself.

But how can I prove a function is actually analytic? Take sin(x) or cos(x), for example. We can easily calculate each Taylor series Tsin(x):=∑

∞

k=0

(−1)kx2k+1

(2k+1)!

and Tcos(x):=∑

∞

k=0

(−1)kx2k

(2k)!

and check the convergence of both. But how do we prove that Tsin(x)=sin(x) and Tcos(x)=cos(x) for all x∈R?

What about other examples (tan(x), ex etc)? Do we really have to treat each case separately? Is there any theorem that makes this task easier?

Thanks!