# Taylor series and Maclaurin series problems

I'm currently working on these two problems, and I'm getting really confused with them. Can someone walk me through them?

1. Find the Maclaurin Series for $$f(x)=\cos\left(\sqrt x\right)$$ and use it to evaluate $$\int\cos\left(\sqrt x\right)\mathrm dx$$ as a series.

2. Find the Taylor Series for $$f(x)=\ln(2-x)$$ about $$x=-1$$.

• imgur.com/0zqQ4IK Apr 21, 2015 at 14:08
• Here is a tutorial to type your question with Latex. Apr 21, 2015 at 14:13

Hint:

$1)$ $$\cos (x ^{1/2}) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n/2}}{(2n)!} = \sum_{n=0}^{\infty} (-1)^n \frac{x^{n}}{(2n)!}$$

If $[a,b] \subset (-1,1)$ then you may use termwise integration.

$2)$ $$\ln (2 - x) = \ln (3 - 1 - x) = \ln (3 - (x + 1)) = \ln 3 + \ln \Big(1 -\frac{1}{3}(x + 1) \Big)$$

• Was my work for the taylor series question correct up to the point I got to? Apr 21, 2015 at 14:17
• Please, write in Latex, it's way better. Apr 21, 2015 at 14:24
• Do you know the series of $\ln (1 - x)$? And $\cos x$? Apr 21, 2015 at 14:27
• Im not sure how to use latex sorry. I do not know the series of the two. Apr 21, 2015 at 14:30
• I've just linked to a tutorial to use latex. And it would be very helpful if you knew them. It's not difficult to derive them, at all. You may check here and here. Apr 21, 2015 at 14:36