**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.