15. Power Series

Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.

tan ⁻¹ (1/2)

Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.

cos 2

{Use of Tech} Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.

∫₀⁰ᐧ³⁵ tan ⁻¹x dx

{Use of Tech} Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.

∫₀⁰ᐧ² (ln (1 + t))/t dt

{Use of Tech} Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.

∫₀⁰ᐧ² sin x² dx

Power series for derivatives

a. Differentiate the Taylor series centered at 0 for the following functions.

b. Identify the function represented by the differentiated series.

c. Give the interval of convergence of the power series for the derivative.

f(x) = eˣ

Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.

ln (1 + x²)

Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.

1/(1 − 2x)

Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.

(1 + x⁴)⁻¹

Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.

sinh x²

Limits by power series Use Taylor series to evaluate the following limits.

lim ₙ → 0 (x²/2 - 1 + cos x)/x⁴

Limits by power series Use Taylor series to evaluate the following limits.

lim ₙ → ₄ ln (x - 3)/(x² - 16)

Suppose you use a second-order Taylor polynomial centered at 0 to approximate a function f. What matching conditions are satisfied by the polynomial?

Does the accuracy of an approximation given by a Taylor polynomial generally increase or decrease with the order of the approximation? Explain.

The first three Taylor polynomials for f(x)=√(1+x) centered at 0 are p₀ = 1, p₁ = 1+x/2, and p₂ = 1 + x/2 − x²/8. Find three approximations to √1.1.