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 $x=-1$.



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

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

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

  • $\begingroup$ Was my work for the taylor series question correct up to the point I got to? $\endgroup$ – user231507 Apr 21 '15 at 14:17
  • $\begingroup$ Please, write in Latex, it's way better. $\endgroup$ – Aaron Maroja Apr 21 '15 at 14:24
  • $\begingroup$ Do you know the series of $\ln (1 - x)$? And $\cos x$? $\endgroup$ – Aaron Maroja Apr 21 '15 at 14:27
  • $\begingroup$ Im not sure how to use latex sorry. I do not know the series of the two. $\endgroup$ – user231507 Apr 21 '15 at 14:30
  • $\begingroup$ 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. $\endgroup$ – Aaron Maroja Apr 21 '15 at 14:36

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