Let k be a positive constant. (a) Show that ekx > k2x2/2, for x > 0. (b) Deduce that e-kx <2/k2x2, for x > 0. (c) Show that x e-kx approaches 0 as x→∞.
> Find the Taylor series at x = 0 of the given function. Use suitable operations on the Taylor series at x = 0 of 1/(1 – x) and ex. ln(1 + x3)
> Find the Taylor series at x = 0 of the given function. Use suitable operations on the Taylor series at x = 0 of 1/(1 – x) and ex. 1/(1 + x3)
> For what values of p is ∑k=1 ∞ 1/pk convergent?
> For what values of p is ∑k=1 ∞ 1/kp convergent?
> Determine if the given series is convergent. ∑k=0∞ k3/(k4 + 1)2
> Determine if the given series is convergent. ∑k=1∞ (ln k)/k
> The third remainder for f (x) at x = 0 is R3 (x) = f (4) (c)/4! x4, where c is a number between 0 and x. Let f (x) = cos x, as in Check Your Understanding Problem 11.1. (a) Find a number M such that | f (4)(c) | ≤ M for all values of c. (b) In Check Your
> Determine if the given series is convergent. ∑k=1∞ 1/3k
> Determine if the given series is convergent. ∑k=1∞ 1/k3
> Find ∑k=0∞ (3k + 5k)/7k.
> Use properties of convergent series to find a ∑k=0∞ (1 + 2k)/3k.
> Find the sum of the given infinite series if it is convergent. 1 + 1/3 + 1/2! (1/3)2 + 1/3! (1/3)3 + 1/4! (1/3)4 + …
> Find the sum of the given infinite series if it is convergent. 1 + 2 + 22/2! + 23/3! + 24/4! + …
> Find the sum of the given infinite series if it is convergent. 1/m – 1/m2 + 1/m3 – 1/m4 + 1/m5 - …, where m is a positive number
> Find the sum of the given infinite series if it is convergent. 1/(m + 1) + m/(m + 1)2 + m2/(m + 1)3 + m3/(m + 1)4 +…, where m is a positive number
> Find the sum of the given infinite series if it is convergent. 22/7 - 25/72 + 28/73 - 211/74 + 214/75 - …
> Find the sum of the given infinite series if it is convergent. 1/8 + 1/82 + 1/83 + 1/84 + 1/85 + …
> If f (x) = 2 - 6 (x - 1) + 3/2! (x - 1)2 – 5/3! (x - 1)3 + 1/4! (x - 1)4, what are f ’’(1) and f ’’’(1)?
> Find the sum of the given infinite series if it is convergent. 52/6 + 53/62 + 54/63 + 55/64 + 56/65 + …
> Find the sum of the given infinite series if it is convergent. 1 – ¾ + 9/16 – 27/64 + 81/256 - …
> Use the Newton–Raphson algorithm with n = 3 to approximate the solution of the equation e2x = 1 + e-x.
> Use the Newton–Raphson algorithm with n = 2 to approximate the zero of x2 - 3x - 2 near x0 = 4.
> (a) Use the third Taylor polynomial of ln(1 - x) at x = 0 to approximate ln 1.3 to four decimal places. (b) Find an approximate solution of the equation ex = 1.3 using the Newton–Raphson algorithm with n = 2 and x0 = 0. Express your answer to four decima
> (a) Find the second Taylor polynomial of √x at x = 9. (b) Use part (a) to estimate √8.7 to six decimal places. (c) Use the Newton–Raphson algorithm with n = 2 and x0 = 3 to approximate the solution of the equation x2 - 8.7 = 0. Express your answer to six
> Use a second Taylor polynomial at x = 0 to estimate the value of tan(.1).
> Use a second Taylor polynomial at t = 0 to estimate the area under the graph of y = -ln(cos 2t) between t = 0 and t = 1/2.
> Find the third Taylor polynomial of ex at x = 2.
> Find the third Taylor polynomial of x2 at x = 3.
> If f (x) = 3 + 4x – 5/2! x2 + 7/3! x3, what are f ’’(0) and f ’’’(0)?
> 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→∞.
> (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
> 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?
