Exam 10: Approximating Functions Using Series

Find the first four terms of the Taylor series about x = -2 for $\ln (1-x)$ .

Use the derivative of the Taylor series about 0 for $\frac{1}{1-x}$ to find the Taylor series about 0 for $\frac{x}{(1-x)^{2}}$ .Use this result to find the value of $\frac{1}{4}+\frac{2}{16}+\frac{3}{64}+\frac{4}{256}+\frac{5}{1024}+\cdots$ .Round to 3 decimal places.

Find the number to which the series $1-\frac{5}{2 !}+\frac{25}{4 !}-\frac{125}{6 !}+\cdots$ converges.Round to 5 decimal places.

Find the 12th-degree Taylor polynomial for $x \sin \left(x^{2}\right)$ centered at x = 0.Suppose you use the first two non-zero terms of the series to approximate $x \sin \left(x^{2}\right)$ for 0 < x < 1.Is the magnitude of the error always less than 0.011?

Find the fourth term of the Taylor series for the function $f(x)=\ln (x+1)$ about x = 1.

Approximate $\int_{0}^{0.3} \sqrt{1+x} d x$ using the first three terms of the Taylor series about zero for $(1+x)^{1 / 2}$ .

Based on the Maclaurin series for the function $f(x)=1-\cos x$ , evaluate $\lim _{x \rightarrow 0} \frac{1-\cos x}{2 x^{2}}$ .

Approximate the function $f(x)=x^{3} e^{-x^{2}}$ for values of x near 0 using the first three non-zero terms of its Taylor polynomial.

Medicine balls are launched from the floor to a height of six feet.They bounce, reaching x/10 the height of the previous bounce each time.The heavier the medicine ball, the smaller the value of x.Write a power series that gives the total distance that a medicine ball bounces as a function of x.What is the function that gives this Taylor polynomial?

What is the interval of convergence of the Taylor series for the function $f(x)=9(1+x)^{1 / 2}$ about zero? (Exclude any possible endpoints.)

Solve $1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\cdots=e^{4}$ for x.