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

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Homework Help Overview

The discussion revolves around finding the Taylor series generated by the function e^(x^2) centered at 0, as well as expressing the integral of e^(x^2) as a Taylor series. The subject area includes series expansions and calculus.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to substitute x^2 into the Taylor series for e^x and then integrate that series for part b. Some participants question the correctness of the approach taken for part b, suggesting that a derivative was mistakenly calculated instead of an integral.

Discussion Status

The discussion is ongoing, with participants providing feedback on the original poster's attempts. There is recognition of a mistake regarding the differentiation and integration process, and a suggestion to include the constant of integration has been made.

Contextual Notes

Participants are navigating the complexities of Taylor series and integration, with some assumptions about the correctness of their methods being questioned. The original poster's approach may be constrained by their understanding of the series and integration techniques.

Lo.Lee.Ta.
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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! :)
 
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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:
 
#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! :)
 
Don't forget the constant of integration ... Otherwise, it looks fine.
 

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