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

[Lo.Lee.Ta.]

1.
a. Find Taylor series generated by e^x2 centered at 0.

b. Express ∫e^x2dx 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:

e^x: 1 + x + x²/2! + x³/3! + ...

e^x2: 1 + x² + x⁴/2! + x⁶/3! + ...


For part b, I just took the integral of the Taylor series for e^x2:

= 0 + 2x + 1/2*4x³ + 1/6*6x⁵ + ...

= 2x + 2x³ + x⁵ + ...

Is this the right way to go about this?
Thanks! :)

 

[tiny-tim]

HI Lo.Lee.Ta.!

a. looks fine

For part b, I just took the integral …

Looks like the derivative to me

 

[Lo.Lee.Ta.]

#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! :)

 

[Mark44]

Don't forget the constant of integration ... Otherwise, it looks fine.