Recitation 26

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: $T_3(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2 + f^{(3)}(a)(x-a)^3/6$.