Define $f(x)=\sqrt{1+x}$ for all $x\in(1,\infty).$ Prove that the Taylor series converges to $f$ for all $x\in(0,1)$.

Question

Define $$f(x)=\sqrt{1+x}$$ for all $$x\in(1,\infty).$$

Prove that the Taylor series converges to $f$ for all $x\in(0,1)$.

I have no idea how to prove this. Can somebody please help me. Thanks in advance.

Answer 1

Recognizing that $n$th coefficient in the Taylor series of $\sqrt{x+1}$(a binomial with power $\frac{1}{2}$) is given by $a_n=\frac{\prod_{n=0}^{\infty}{(\frac{1}{2}-n)}}{n!}$. You can use ratio test to find that the series must converge absolutely for $|x|<1$ because $\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|<1$ for $|x|<1$

Then convergence for $|x|<1$ implies that the series is also convergent for $x\in(0,1)$