Finding Sum of First Terms

Question

Hello I was wondering if I could get some help trying to find a sum of the first three terms in series.

The problem statement is:

"Write the first three terms in the series for $y(t) = e^{t^2}$, and us it to approximate y(2)"

I know that I need to use the Taylor series to calculate this, but for some reason when I try to use it I am struggling with the $t^2$ part.

Thanks for the help!

Answer 1

$$e^t=1+t+\frac{t^2}{2!}+\frac{t^3}{3!}+...\to \\ e^{t^2}=1+t^2+\frac{t^4}{2!}+\frac{t^6}{3!}+...\approx 1+t^2+\frac{t^4}{2}\to y(2)\approx 13$$

Answer 2

Let $$x= t^2$$ in the Taylor expansion of $e^x$ to get $$e^{t^2}=1+t^2+\frac{t^4}{2!}+\frac{t^6}{3!}+...$$ The three term approximation is $13$. This approximation is very weak, because we are far away from the center, $t=0$.

The error is $e^4 - 13 = 41.59$ which is not good at all.