Section 11.10 C

Key Problems: 47, 48, 41, 59, 68, 73, 82, 86

Problem 47

$$f(x)={\sin }^{2}x$$

Hint: Use $si{n}^{2}x=\frac{1}{2}(1-{\cos }_{2}x)$.

Problem 48

$$f(x)=\{\begin{array}{l}\frac{x-\sin x}{x^3}\text{ if }x\ne 0\\ \frac{1}{6}\text{ if }x=0\end{array}$$

Problem 41

Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function.

$$f(x)=x\cos 2x$$

Problem 59

Evaluate the indefinite integral as an infinite series.

$$\int \sqrt{1+x^3}dx$$

Problem 68

Use series to evaluate the limit.

$$\underset{x\to 0}{\lim }\frac{1-\cos x}{1+x-e^x}$$

Problem 73

Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function.

$$y=e^{-x^2}\cos x$$

Problem 82

Find the function represented by the given power series.

$$\sum _{n=0}^{\infty }(-1{)}^n\frac{x^{2n+1}}{2^{2n+1}(2n+1)!}$$

Problem 86

Find the sum of the series.

$$\sum _{n=0}^{\infty }\frac{3^n}{5^{n}n!}$$