Yesterday I have found and old homework with an exercise that I couldn't do but ironically it's so difficult too right now. Is the next:
Let $f(x)=(1-x^{2})^{1/2}$. Find a power series of the form $\displaystyle\sum_{n=0}^{\infty}c_n x^{n}$ such that converges uniformly to $f(x)$ in the interval $[0,1]$.
My first thought was Taylor series of this function but I don't think that that will be a good idea. Then, searching in StackExchange I have found this answer. I thik that is the right way, but, prove the uniformly convergence for my particular series is so difficult. Moreover, after read the Wikipedia article that the user provides, the proof of the convergence of the series are so tecnic and it's so away from my knowledge. Is there another form to prove that the series converges? Or, moreover, is there another form to obtain a power series such that converges uniformly to $f(x)$ in the interval $[0,1]$?. Even, if I can prove that the series converges, how can I prove that the series converges to $f(x)$? Maybe prove the convergence it's not difficult using the $M$-test of Weierstrass (A first idea suggested by Wikipedia article). I really aprecciate any help you can provide me. Thanks!
Best Answer
My 2 cents: On interval $[0,1]$ Taylor series for function $f(x)=(1-x^{2})^{1/2}$ $$\sum_{k=0}^{\infty}\binom{\frac{1}{2}}{k}(-x^2)^k\ \ \ \ (1)$$ can be estimated by series $$\sum{}{}\frac{|\alpha(\alpha-1)\ldots(\alpha-k+1)|}{k!}$$ convergence for last series comes from Raabe's_test for $\alpha > 0$. And then Weierstrass gives uniformly convergence for (1).
Convergence of series (1) on $[0,1]$ to function $f$ is well known fact (for example Spivak "Calculus" IV edition, 495p i.e. *21 from 23), so we can conclude, that (1) is desired series, which converges uniformly to $f(x)$ in the interval $[0,1]$.