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How do you find the power series representation for the function #f(x)=e^(x^2)# ?

Answer 1

Emma Aldridge

Since

#e^x=sum_{n=0}^infty{x^n}/{n!}#,

by replacing #x# by #x^2#,

#e^{x^2}=sum_{n=0}^infty{(x^2)^n}/{n!}=sum_{n=0}^infty{x^{2n}}/{n!}#.

I hope that this was helpful.

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Answer 2

Ezra Adams

To find the power series representation for the function ( f(x) = e^{x^2} ), we can use the Maclaurin series expansion for ( e^x ):

[ e^x = \sum_{n=0}^\infty \frac{x^n}{n!} ]

Replacing ( x ) with ( x^2 ) in the series expansion, we get:

[ e^{x^2} = \sum_{n=0}^\infty \frac{(x^2)^n}{n!} ]

[ = \sum_{n=0}^\infty \frac{x^{2n}}{n!} ]

So, the power series representation for ( f(x) = e^{x^2} ) is:

[ f(x) = \sum_{n=0}^\infty \frac{x^{2n}}{n!} ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.