15. Power Series

Taylor Series & Taylor Polynomials

15. Power Series

Taylor Series & Taylor Polynomials

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Multiple Choice

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

$open bracket negative 1 comma 1 close bracket$

$open paren negative 1 comma 1 close bracket$

$(- \sqrt{3}, \sqrt{3})$

$[- \sqrt{3}, \sqrt{3})$

Verified step by step guidance

Step 1: Recall that the Maclaurin series is a Taylor series centered at x = 0. To find the interval of convergence, we need to determine the values of x for which the series converges.

Step 2: Write the Maclaurin series for f(x) = tan⁡−1(x). The series is given by: f(x) = x - x³/3 + x⁵/5 - x⁷/7 + ... This is an alternating series with terms decreasing in magnitude.

Step 3: Apply the ratio test to determine the interval of convergence. The ratio test states that a series Σaₙ converges absolutely if lim (n→∞) |aₙ₊₁/aₙ| < 1. Compute the ratio of consecutive terms for the series.

Step 4: Solve the inequality obtained from the ratio test to find the range of x values for which the series converges. This will give the interval of convergence.

Step 5: Check the endpoints of the interval separately to determine whether the series converges at x = -1 and x = 1. Use the alternating series test or direct substitution to verify convergence at these points.

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Textbook Question

{Use of Tech} Approximating sin x Let f(x)=sin x, and let pₙ and qₙ be nth−order Taylor polynomials for f centered at 0 and π, respectively.

a. Find p₅ and q₅

b. Graph f, p₅, and q₅ on the interval [−π, 2π]. On what interval is p₅ a better approximation to f than q₅? On what interval is q₅ a better approximation to f than p₅?

c. Complete the following table showing the errors in the approximations given by p₅ and q₅ at selected points.

d. At which points in the table is p₅ a better approximation to f than q₅? At which points do p₅ and q₅ give equal approximations to f? Explain your observations.

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Textbook Question

{Use of Tech} A different kind of approximation When approximating a function f using a Taylor polynomial, we use information about f and its derivative at one point. An alternative approach (called interpolation) uses information about f at several different points. Suppose we wish to approximate f(x)=sin x on the interval [0, π].

a. Write the (quadratic) Taylor polynomial p₂ for f centered at π/2.

b. Now consider a quadratic interpolating polynomial q(x) = ax² + bx + c. The coefficients a, b, and c are chosen such that the following conditions are satisfied:

q(0) = f(0), q(π/2) = f(π/2), and q(π) = f(π)

Show that q(x) = −(4/π²)x² + (4/π)x.

c. Graph f, p₂, and q on [0, π].

d. Find the error in approximating f(x) = sin x at the points π/4, π/2, 3π/4, and π using p₂ and q.

e. Which function, p₂ or q, is a better approximation to f on [0, π]? Explain.

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Multiple Choice

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

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Multiple Choice

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

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Multiple Choice

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

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Multiple Choice

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

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Multiple Choice

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

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Multiple Choice

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

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Struggling with Calculus?

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

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Taylor Series & Taylor Polynomials practice set

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