# Thread: Taylor and Maclaurin series

1. ## Taylor and Maclaurin series

Find the Taylor and Maclaurin series of the given function with the given point $\displaystyle z_0$ as center and determine the radius of convergence.

$\displaystyle e^{\frac{z^2}{2}}\int e^{-t^2} dt$

By the way, integrate from 0 to z and center at $\displaystyle z_0 = 0$

I cannot imagine, how can I solve this.

2. Originally Posted by noppawit
Find the Taylor and Maclaurin series of the given function with the given point $\displaystyle z_0$ as center and determine the radius of convergence.

$\displaystyle e^{\frac{z^2}{2}}\int e^{-t^2} dt$

By the way, integrate from 0 to z and center at $\displaystyle z_0 = 0$

I cannot imagine, how can I solve this.
centered at 0 ...

$\displaystyle e^u = 1 + u + \frac{u^2}{2!} + \frac{u^3}{3!} + ...$

$\displaystyle e^{-t^2} = 1 - t^2 + \frac{t^4}{2!} - \frac{t^6}{3!} + ...$

$\displaystyle \int_0^z e^{-t^2} \, dt = \left[t - \frac{t^3}{3} + \frac{t^5}{5 \cdot 2!} - \frac{t^7}{7 \cdot 3!} + ... \right]_0^z = z - \frac{z^3}{3} + \frac{z^5}{5 \cdot 2!} - \frac{z^7}{7 \cdot 3!} + ...$