1. ## MacLaurin Seires

Hello,
I am trying to find the Maclaurin series (Taylor series centered about x=0) as well as the radius of converge for
$f(x)=\frac{1}{\sqrt{1+x}}$

Also, assuming the above series represent f(x) on the above interval of convergence, can that series be used to find a Maclaurin series for
$\frac{1}{\sqrt{1-t^2}}$? (if it can, how?)

Am I on the right track with the second part? I am a bit stalled on part a).

Thank you for any help or direction

2. Given a real number $\alpha$ , the function $f(x) = (1+x)^{\alpha}$ can be written as McLaurin series for $-1 < x < 1$ as follows...

$(1+x)^{\alpha}= \sum_{n=0}^{\infty} \binom {\alpha}{n}\cdot x^{n}$ (1)

... where...

$\binom {\alpha}{n} = \frac{\alpha\cdot (\alpha-1)\cdot (\alpha-2)\dots (\alpha-n+1)}{n!}$ (2)

Such series is called binomial series. Setting in (1) $\alpha= - \frac{1}{2}$ we have...

$\frac{1}{\sqrt{1+x}}= 1 - \frac{1}{2}\cdot x + \frac{1\cdot 3}{2\cdot 4}\cdot x^{2} - \frac{1\cdot 3 \cdot 5}{2\cdot 4\cdot 6}\cdot x^{3} + \dots + (-1)^{n} \frac{1\cdot 3\cdot 5\dots (2n-1)}{2\cdot 4\cdot 6\dots 2n}\cdot x^{n} +\dots$ (3)

If You have to write the McLaurin expansion of $f(t)= \frac{1}{\sqrt{1-t^{2}}}$, all what you have to do is to write $-t^{2}$ instead of $x$ in (3)...

Kind regards

$\chi$ $\sigma$