15. Power Series

Taylor series and interval of convergence

c. Determine the interval of convergence of the series.

f(x)=3ˣ, a=0

Find a Taylor series for f centered at 2 given that f⁽ᵏ⁾(2)=1, for all nonnegative integers k.

Use of Tech Linear and quadratic approximation

a. Find the linear approximating polynomial for the following functions centered at the given point a.

b. Find the quadratic approximating polynomial for the following functions centered at a.

c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.

f(x) = 8x^(3/2), a=1; approximate 8 ⋅ 1.1^(3/2)

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

e. The Taylor series for an even function centered at 0 has only even powers of x.

Taylor series and interval of convergence

a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.

f(x)=3ˣ, a=0

Taylor series

b. Write the power series using summation notation.

f(x) = 2ˣ, a = 1

Taylor series

b. Write the power series using summation notation.

f(x) = 1/x, a = 1

Find the Taylor Series of $f\left(x\right)=\cos x$ centered $x=\pi$. Then, write the power series using summation notation.

Find the interval of convergence for the Taylor series for $f\left(x\right)=\sin x$ centered at $x=\frac{\pi }{2}$.

Find the interval of convergence for the Maclaurin series for $f\left(x\right)={\tan }^{-1}x$

Find the Taylor polynomials of order $0,1,2$, and $3$ for $f\left(x\right)=\ln \left(x\right)$ centered at $x=1$.

Approximate $\ln 1.5$ to four decimal places using the third-degree Taylor polynomial for $f\left(x\right)=\ln x$.

Find the Maclaurin polynomials of order $0,1,2$, and $3$ for $f\left(x\right)=\sin x$

Approximate $\sin 0.3$ to four decimal places using the third-degree Maclaurin polynomial for $f\left(x\right)=\sin x$.

Limits Evaluate the following limits using Taylor series.

lim ₓ→∞ x(e¹/ˣ − 1)