15. Power Series

Use series to evaluate the limits in Exercises 29–40.

29. lim (x → 0) (e^x - (1 + x)) / x²

{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 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.

Find the Taylor polynomial p₃ centered at a=e for f(x)=ln x.

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

Power series from the geometric series Use the geometric series a Σₖ ₌ ₀ ∞ (x)ᵏ = 1/(1 - x), for |x| < 1, to determine the Maclaurin series and the interval of convergence for the following functions.

ƒ(x) = 1/(1 - x²)

Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series

(1 + x)⁻² = 1 − 2x + 3x² − 4x³ + ⋯, for −1 < x < 1.

1/(3 + 4x)²

Tangent line is p₁ Let f be differentiable at x=a

a. Find the equation of the line tangent to the curve y=f(x) at (a, f(a)).

b. Verify that the Taylor polynomial p_1 centered at a describes the tangent line found in part (a).

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)

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) = ln (1 + x)

{Use of Tech} Maximum error Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.

√(1+x) ≈ 1 + x/2 on [−0.1,0.1]

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

lim ₙ → 0 (x²/2 - 1 + cos x)/x⁴

The series

sec x = 1 + x²/2 + 5x⁴/24 + 61x⁶/720 + 277x⁸/8064 + ⋯

converges to sec x for −π/2 < x < π/2.

a. Find the first five terms of a power series for the function ln|sec x + tan x|. For what values of x should the series converge?

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. There is more than one way to choose the center of the series.

sinh (-1)

L'Hôpital's Rule by Taylor series Suppose f and g have Taylor series about the point a.

a. If f(a) = g(a) = 0 and g′(a) ≠ 0, evaluate lim ₓ→ₐ f(x)/g(x) by expanding f and g in their Taylor series. Show that the result is consistent withl’Hôpital’s Rule.

b. If f(a) = g(a) =f′(a) = g′(a) = 0 and g′′(a) ≠ 0, evaluate lim ₓ→ₐ f(x)/g(x) by expanding f and g in their Taylor series. Show that the result is consistent with two applications of 1'Hôpital's Rule.

{Use of Tech} Newton's derivation of the sine and arcsine series Newton discovered the binomial series and then used it ingeniously to obtain many more results. Here is a case in point.

a. Referring to the figure, show that x = sin s or s = sin ⁻¹ x.

b. The area of a circular sector of radius r subtended by an angle θ is 1/2r²θ. Show that the area of the circular sector APE is s/2, which implies that

s = 2 ∫₀ˣ √(1 − t²) dt − x √(1 −x²)

c. Use the binomial series for f(x) = √(1 − x²) to obtain the first few terms of the Taylor series for s=sin ⁻¹ x.

d. Newton next inverted the series in part (c) to obtain the Taylor series for x=sin s. He did this by assuming sin s = ∑ aₖ sᵏ and solving x = sin(sin ⁻¹ x) for the coefficients aₖ. Find the first few terms of the Taylor series for sin s using this idea (a computer algebra system might be helpful as well).