Example 1: Use series to evaluate the limit. (a) \lim_{x\to 0}\frac{x-x^2/2-\ln(1+x)}{x^3}; (b) \lim_{x\to 0}\frac{1-x^/2-\cos x}{1+x+x^2/2+x^3/6-e^x}; (c) \lim_{x\to 0}\frac{\sin x-x+x^3/6-x^5/120}{x^7}.
Hint: Use Maclaurin series of \ln(1+x), \cos x, e^x and \sin x.
Example 2: Find the sum of the series. (a) \sum_{n=1}^\infty(-1)^{n-1}\frac{3^n}{n5^n}; (b) \sum_{n=0}^\infty\frac{(-1)^n\pi^{2n+1}}{4^{2n+1}(2n+1)!}; (c) 3+\frac{9}{2!}+\frac{27}{3!}+\frac{81}{4!}+\ldots.
Hint: (a) Use Maclaurin series of \ln(1+x); (b) Use Maclaurin series of \sin x; (c) Use Maclaurin series of e^x-1;
Example 3: Find the first three nonzero terms in the Maclaurin series for (a) e^x\sin x and (b) \tan x.
Hint: (a) Use Maclaurin series for e^x and \sin x and multiply them together. (b) Use Maclaurin series for \sin x and \cos x and use a procedure like a long division.
Example 4: Find the Taylor polynomial T_3(x) for the function f centered at the number a. (a) f(x) = \cos x, a = \pi /2; (b) f(x)=xe^{-2x}, a=0.
Hint: T_3(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2 + f^{(3)}(a)(x-a)^3/6.