Find Taylor series generated by e^x centered at 0.

In summary, the conversation discusses finding the Taylor series generated by ex2 centered at 0 and expressing ∫ex2dx as a Taylor series. The first part involves substituting x2 for x in the general form of the e^x Taylor series, while the second part involves taking the integral of the Taylor series for ex2. The conversation also includes a correction to the derivative and a reminder to include a constant of integration.
  • #1
Lo.Lee.Ta.
217
0
1.
a. Find Taylor series generated by ex2 centered at 0.

b. Express ∫ex2dx as a Taylor series.

2. For part a, I just put the value of "x2" in place of x in the general form for the e^x Taylor series:

ex: 1 + x + x2/2! + x3/3! + ...

ex2: 1 + x2 + x4/2! + x6/3! + ...


For part b, I just took the integral of the Taylor series for ex2:

= 0 + 2x + 1/2*4x3 + 1/6*6x5 + ...

= 2x + 2x3 + x5 + ...

Is this the right way to go about this?
Thanks! :)
 
Physics news on Phys.org
  • #2
HI Lo.Lee.Ta.! :smile:

a. looks fine
Lo.Lee.Ta. said:
For part b, I just took the integral …

Looks like the derivative to me :redface:
 
  • #3
#O_O AGH! I did take the derivative! Thanks! ha

So, it should be: x + 1/3(x3) + 1/10(x5) + 1/42(x7) + ...

Is this right?

Thanks! :)
 
  • #4
Don't forget the constant of integration ... Otherwise, it looks fine.
 

Related to Find Taylor series generated by e^x centered at 0.

1. What is a Taylor series?

A Taylor series is a mathematical representation of a function as an infinite sum of terms. It is used to approximate a function at a specific point or interval by using derivatives of the function at that point.

2. How is a Taylor series generated?

A Taylor series is generated by evaluating the function and its derivatives at a specific point, known as the center of the series. The coefficients of each term in the series are determined by the value of the derivatives at the center.

3. What is e^x?

e^x is the exponential function where e is the base of the natural logarithm (approximately 2.71828) and x is the exponent. It is commonly used in mathematics, science, and engineering to model growth and decay.

4. What does it mean to center a Taylor series at 0?

To center a Taylor series at 0 means to choose 0 as the center point of the series. This means that the function and its derivatives will be evaluated at x=0 to determine the coefficients of each term in the series.

5. Why is it useful to find the Taylor series generated by e^x centered at 0?

Finding the Taylor series generated by e^x centered at 0 allows us to approximate the value of e^x at any point by using a finite number of terms in the series. This is useful because it can help us solve problems and make predictions in various fields such as physics, chemistry, and economics.

Similar threads

Bounds of the remainder of a Taylor series
    • Calculus and Beyond Homework Help
Replies
27
Views
2K
Evaluate the Taylor series and find the error at a given point
    • Calculus and Beyond Homework Help
Replies
6
Views
2K
Find the Taylor series of a function
    • Calculus and Beyond Homework Help
Replies
10
Views
1K
Find the limit using taylor series
    • Calculus and Beyond Homework Help
Replies
11
Views
2K
Taylor Series Expansion of f(x) at 0
    • Calculus and Beyond Homework Help
Replies
7
Views
986
Taylor Series for f(x) = (1+x)^m
    • Calculus and Beyond Homework Help
Replies
1
Views
2K
B Can We Simplify the Taylor Series Expansion for e^(f(x,y))?
    • General Math
Replies
3
Views
845
Find Taylor Series from a function and its interval of convergence
    • Calculus and Beyond Homework Help
Replies
10
Views
2K
How to show that these sequences are summable?
    • Calculus and Beyond Homework Help
Replies
1
Views
375
How can the Taylor expansion of x^x at x=1 be simplified to make solving easier?
    • Calculus and Beyond Homework Help
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top