Explanation Given: The function f ( x ) = e 2 x . Result used: (1) Ratio test : Let ∑ k = 1 ∞ a k be an infinite series and let r = lim k → ∞ | a k + 1 a k | . If r < 1 , then the series ∑ k = 1 ∞ a k is converges. Calculation: Obtain the power series for the function f ( x ) = e − x . Substitute 2 x for x in e x = ∑ k = 0 ∞ x k k ! , e 2 x = ∑ k = 0 ∞ ( 2 x ) k k ! Hence, the power series representation of e 2 x is ∑ k = 0 ∞ ( 2 x ) k k ! . The interval of convergence of ∑ k = 0 ∞ ( 2 x ) k k ! is computed as follows, Let a k = ( 2 x ) k k ! . Then, a k + 1 = ( 2 x ) k + 1 ( k + 1 ) ! . Use the Ratio test to compute the value | a k + 1 a k |

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 9.2, Problem 70E

To determine

To find: The power series for the given function and interval of convergence by using the exponential function centered at 0 is ex=∑k=0∞xkk!,for−∞<x<∞.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.2, Problem 69E

Chapter 9.2, Problem 71E

Students have asked these similar questions

What are the correct answers for the second and third question on this page. I am on the Cartesian vectors unit in calculu

Trolley of the overhead crane moves along the bridge rail. The trolley position is measured from the center of the bridge rail (x = 0) is given by x(t) = 0.5t^3-6t^2+19.5t-14 : 0 <= t <= 3 min. The trolley moves from point A to B in the forward direction, B to C in the reverse direction and C to D again in the forward direction. CONTROL PANEL END TRUCK- RUNWAY BEAM- BRIDGE RAIL HOIST -TROLLEY TROLLEY BUMPER TROLLEY DRIVE LPENDANT TRACK -TROLLEY CONDUCTOR TRACK WIRE ROPE -HOOK BLOCK -BRIDGE DRIVE -END TRUCK BUMPER -RUNWAY RAIL TROLLEY END STOP -CONDUCTOR BAR PENDANT FESTOONING TROLLEY FESTOONING PENDANT CABLE PENDANT x(t)=0.5t^3-6t^2+19.5t-14 v(t)=1.5t^2-12t+19.5 a(t)=(dv(t))/dt=3t-12 Fig. T2.2: The overhead crane Total masses of the trolley, hook block, and the load attached to the hook block are 110 kg, 20 kg, and 150 kg. Damping coefficient, D, is 40 kg/s. What is the total amount of energy required from the trolley motor to move the system [Hint: Use Newton's 2nd law to obtain the…

CONTROL PANEL- BRIDGE RAIL HOIST -TROLLEY TROLLEY BUMPER -BRIDGE DRIVE END TRUCK- RUNWAY BEAM- END TRUCK BUMPER -RUNWAY RAIL TROLLEY DRIVE TROLLEY END STOP -CONDUCTOR BAR LPENDANT TRACK TROLLEY CONDUCTOR TRACK -WIRE ROPE PENDANT FESTOONING TROLLEY FESTOONING -PENDANT CABLE -HOOK BLOCK PENDANT

Chapter 9 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 9.1 - Suppose you use a second-order Taylor polynomial...Ch. 9.1 - Does the accuracy of an approximation given by a...Ch. 9.1 - The first three Taylor polynomials for f(x)=1+x...Ch. 9.1 - Prob. 4E Ch. 9.1 - How is the remainder Rn(x) in a Taylor polynomial...Ch. 9.1 - Explain how to estimate the remainder in an...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The power series for the given function and interval of convergence by using the exponential function centered at 0 is e x = ∑ k = 0 ∞ x k k ! , for − ∞ < x < ∞ .

Solution Summary: The author explains how to find the power series for the function f(x)=e2x

Videos

To find: The power series for the given function and interval of convergence by using the exponential function centered at 0 is ex=∑k=0∞xkk!,for−∞<x<∞.

Want to see the full answer?

Chapter 9 Solutions

The power series for the given function and interval of convergence by using the exponential function centered at 0 is e x = ∑ k = 0 ∞ x k k ! , for − ∞ &lt; x &lt; ∞ .

Solution Summary: The author explains how to find the power series for the function f(x)=e2x

Videos

To find: The power series for the given function and interval of convergence by using the exponential function centered at 0 is ex=∑k=0∞xkk!,for−∞<x<∞.

Want to see the full answer?

Chapter 9 Solutions

The power series for the given function and interval of convergence by using the exponential function centered at 0 is e x = ∑ k = 0 ∞ x k k ! , for − ∞ < x < ∞ .