Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Problem 2. ln(1 + x2)
> Determine the third Taylor polynomial of the given function at x = 0. f (x) = xe3x
> Find the nth Taylor polynomial of 2/(2 – x) at x = 0.
> In what way is the nth Taylor polynomial of f (x) at x = a like f (x) at x = a?
> Define the nth Taylor polynomial of f (x) at x = a.
> Discuss the three possibilities for the radius of convergence of a Taylor series.
> Define the Taylor series of f (x) at x = 0.
> What is the sum of a convergent geometric series?
> What is a geometric series and when does it converge?
> What is meant by the sum of a convergent infinite series?
> What is a convergent infinite series? Divergent?
> What is the nth partial sum of an infinite series?
> Determine the nth Taylor polynomial of f (x) = 1/x at x = 1.
> Explain how the Newton–Raphson algorithm is used to approximate a zero of a function.
> State the remainder formula for the nth Taylor polynomial of f (x) at x = a.
> Find the Taylor series at x = 0 of the given function by computing three or four derivatives and using the definition of the Taylor series. ln(1 - 3x)
> Find the Taylor series at x = 0 of the given function by computing three or four derivatives and using the definition of the Taylor series. 1/(2x + 3)
> Let Rn(x) be the nth remainder of f (x) = ex at x = 0. (See Section 11.1.) Show that, for any fixed value of x, |Rn(x)| ≤ e|x| * |x |n+1/(n + 1) |, and hence, conclude that |Rn(x)| → 0 as n → ∞. This shows that the Taylor series for ex converges to ex fo
> Let Rn(x) be the nth remainder of f (x) = cos x at x = 0. (See Section 11.1.) Show that, for any fixed value of x, |Rn(x)| ≤ |x|n+1/(n + 1) |, and hence, conclude that |Rn(x)| → 0 as n → ∞. This shows that the Taylor series for cos x converges to cos x f
> If k is a positive constant, show that x2e-kx approaches 0 as x→∞.
> Show that ex > x3/6 for x > 0, and from this, deduce that x2 e-x approaches 0 as x→∞.
> Let k be a positive constant. (a) Show that ekx > k2x2/2, for x > 0. (b) Deduce that e-kx 0. (c) Show that x e-kx approaches 0 as x→∞.
> (a) Use the Taylor series for ex at x = 0 to show that ex > x2/2 for x > 0. (b) Deduce that e-x < 2/x2 for x > 0. (c) Show that xe-x approaches 0 as x → ∞.
> Determine all Taylor polynomials of f (x) = x4 + x + 1 at x = 2.
> Find an infinite series that converges to the value of the given definite integral. ∫0 1 x ex3 dx
> Find an infinite series that converges to the value of the given definite integral. ∫0 1 e-x2 dx
> Find an infinite series that converges to the value of the given definite integral. ∫0 1 sin x2 dx
> Find the Taylor series expansion at x = 0 of the given antiderivative. ∫1/(1 + x3) dx
> Find the Taylor series expansion at x = 0 of the given antiderivative. ∫x ex3 dx
> Find the Taylor series expansion at x = 0 of the given antiderivative. ∫e-x2 dx
> The Taylor series at x = 0 for (1 + x2)/(1 – x) is 1 + x + 2x2 + 2x3 + 2x4 + … . Find f (4)(0), where f (x) = (1 + x4)/(1 - x2).
> The Taylor series at x = 0 for f (x) = tan x is x + 1/3 x3 + 2/15 x5 + 17/315 x7 + … . Find f (4)(0).
> The Taylor series at x = 0 for f (x) = sec x is 1 + ½ x2 + 5/24 x4 + 61/720 x6 + … . Find f (4)(0).
> The Taylor series at x = 0 for f (x) = ln [(1 + x)/(1 – x)] is given in Exercise 22. Find f (5)(0). Exercise 22: Show that ln [(1 + x)/(1 – x)] = 2x + 2/3 x3 + 2/5 x5 + 2/7 x7 + … , | x | < 1. This series converges much more quickly than the series for
> Use the second Taylor polynomial of f (x) = ln x at x = 1 to estimate ln .8.
> Use the Taylor series for cos x (see Problem 2 in Check Your Understanding) to show that cos(-x) = cos x.
> Use the Taylor series for ex to show that d/dx ex = ex.
> Use the Taylor series expansion for x/(1 - x)2 to find the function whose Taylor series is 1 + 4x + 9x2 + 16x3 + 25x4 + … .
> Use Exercise 25 and the fact that ∫ 1/√(1 - x2) dx = ln(x + √(1 + x2)) + C to find the Taylor series of ln(x + √(1 + x2)) at x = 0. Exercise 25: Find the first four terms in the Taylor series of 1/√(1 - x2) at x = 0.
> Find the first four terms in the Taylor series of 1/√(1 - x2) at x = 0.
> Given the Taylor series expansion 1/√(1 + x) = 1 – ½ x + ½ * ¾ x2 - ½ * ¾ * 5/6 x3 + ½ * ¾ * 5/6 * 7/8 x4 - … , find the first four terms in the Taylor series of 1/√(1 - x) at x = 0.
> The hyperbolic sine of x is defined by sinh x = ½ (ex - e-x). Repeat parts (a) and (b) of Exercise 23 for sinh x. Exercise 23: The hyperbolic cosine of x, denoted by cosh x, is defined by cosh x = ½ (ex + e-x). This function occurs often in physics and
> The hyperbolic cosine of x, denoted by cosh x, is defined by cosh x = ½ (ex + e-x). This function occurs often in physics and probability theory. The graph of y = cosh x is called a catenary. (a) Use differentiation and the definition of a Taylor series
> Show that ln [(1 + x)/(1 – x)] = 2x + 2/3 x3 + 2/5 x5 + 2/7 x7 + … , | x | < 1. This series converges much more quickly than the series for ln(1 - x) in Example 3, particularly for x close to zero. The series gives a formula for ln y, where y is any numb
> Find the Taylor series of xex2 at x = 0
> Use the second Taylor polynomial of f (x) = √x at x = 9 to estimate √9.3.
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Determine the third and fourth Taylor polynomials of x3 + 3x - 1 at x = -1.
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function by computing three or four derivatives and using the definition of the Taylor series. (1 + x)3
> Find the Taylor series at x = 0 of the given function by computing three or four derivatives and using the definition of the Taylor series. √(1 + x)
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=1∞ 5/k3/2
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=1∞ 3/√k
> Determine the third and fourth Taylor polynomials of cos x at x = π.
> Use Exercise 30 to show that the series ∑k=1∞ 3/k2 is convergent. Then, use the comparison test to show that the series ∑k=1∞ e1/k/k2 is convergent. Exercise 30: Let ∑k=1∞ ak be a convergent series with sum S, and let c be a constant. Then, ∑k=2∞ cak is
> Use Exercise 29 to show that the series ∑k=0∞ (8k + 9k)/10k is convergent, and determine its sum. Exercise 29: The following property is true for any two series (with possibly some negative terms): Let ∑k=1∞ ak and ∑k=1∞ bk be convergent series whose su
> Let ∑k=1∞ ak be a convergent series with sum S, and let c be a constant. Then, ∑k=2∞ cak is a convergent series whose sum is c * S. Make a geometric picture to illustrate why this is true when c = 2 and the terms ak are all positive.
> The following property is true for any two series (with possibly some negative terms): Let ∑k=1∞ ak and ∑k=1∞ bk be convergent series whose sums are S and T, respectively. Then, ∑k=1∞ (ak + bk) is a convergent series whose sum is S + T. Make a geometric
> Can the comparison test be used with a ∑k=1∞ 1/(k2 ln k) and ∑k=2∞ 1/k2 to deduce anything about the first series?
> Can the comparison test be used with ∑k=2∞ 1/(k ln k) and ∑k=2∞ 1/k to deduce anything about the first series?
> Use the comparison test to determine whether the infinite series is convergent or divergent. ∑k=0∞ 1/(3/4)k + (5/4)k [Compare with ∑k=0∞ (3/4)-k or ∑k=0∞ (5/4)-k.]
> Use the comparison test to determine whether the infinite series is convergent or divergent. ∑k=1∞ 1/5k cos2 (kπ/4) [Compare with ∑k=1∞ 1/5k.]
> Use the comparison test to determine whether the infinite series is convergent or divergent. ∑k=1∞ 1/k3k [Compare with ∑k=1∞ 1/3k.]
> Use the comparison test to determine whether the infinite series is convergent or divergent. ∑k=1∞ 1/(2k + k) [Compare with ∑k=1∞ 1/2k.]
> Determine the fourth Taylor polynomial of ln x at x = 1.
> Use the comparison test to determine whether the infinite series is convergent or divergent. ∑k=2∞ 1/√(k2 – 1) [Compare with ∑k=2∞ 1/k.]
> Use the comparison test to determine whether the infinite series is convergent or divergent. ∑k=2∞ 1/(k2 + 5) [Compare with ∑k=2∞ 1/k2.]
> Is the series a ∑k=1∞ 3k/4k convergent? What is the easiest way to answer this question? Can you tell if ∫1∞3x/4x dx is convergent?
> It can be shown that lim b → ∞ be-b = 0. Use this fact and the integral test to show that a ∑k=1∞ k ek is convergent.
> Use the integral test to determine if a ∑k=1∞ e1/k k2 is convergent. Show that the hypotheses of the integral test are satisfied.
> It can be shown that ∫0∞ 3/(9 + x2) dx is convergent. Use this fact to show that an appropriate infinite series converges. Give the series, and show that the hypotheses of the integral test are satisfied.
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=2∞ (k + 1)/(k2 + 2k + 1)2
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=1∞ (2k + 1)/(k2 + k + 2)
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=1∞ k-3/4
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=1∞ ke-k2
> Determine the third Taylor polynomial of 1/(5 – x) at x = 4.
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=1∞ 1/e2k+1
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=1∞ e3-k
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=1∞ 1/(3k)2
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=2∞ 1/k(ln k)2
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=1∞ 1/(2k + 1)3
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=2∞ k/(k2 + 1)3/2
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=2∞ 1/k√(ln k)
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=1∞ 2/(5k – 1)
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=0∞ 7/(k + 100)
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=2 ∞ 1/(k - 1)3
> Use a second Taylor polynomial at x = 0 to estimate the area under the curve y = √(cos x) from x = -1 to x = 1. (The exact answer to three decimal places is 1.828.)
> Determine the sums of the following geometric series when they are convergent. 1 + 1/6 + 1/62 + 1/63 + 1/64 …
