# Find the Taylor polynomial of degree n

-Find the Taylor polynomial of degree $$n$$ for $$f(x) = \sqrt{(1+x)}$$, centered at $$x = 0$$.

-I know how to find taylor polynomial but when it comes to finding the $$n^{th}$$ term, I'm a little lost on finding what the pattern is.

We have $$f^{'}(0) = \frac{1}{2}, f^{''}(0) = -\frac{1}{4}, f^{'''}(0) = \frac{3}{8}, f^{4}(0) = \frac{-3*5}{16}$$ and $$f^{n}(0) = \frac{-1^{n+1}}{2^n}*$$ ?

Thanks!

• You're right-- it's not an easy pattern-- it has to involve with binomial coefficients. Apr 14, 2020 at 0:24
• Write it as $(1+x)^{1/2}$. Keep differentiating. YOu should see a pattern in terms of binomial coefficients Apr 14, 2020 at 0:24

It would be easier to apply the bionomial theorem. $$(1+x)^a=\sum_{n=0}^\infty\binom{a}{n} x^n$$ and, if required, truncate where ever you need.
If $$a=\frac 12$$, remember that $$\binom{a}{n}=\frac{\sqrt{\pi }}{2 \left(\frac{1}{2}-n\right)! n!}$$ which then makes $$\sqrt{1+x}=\frac{\sqrt{\pi }}{2}\sum_{n=0}^\infty\frac {x^n}{\left(\frac{1}{2}-n\right)!\,n!}$$