Taylor Series/Limits/Convergence Test

Hey everyone,

I am having some trouble working through these problems, and I was hoping that someone can help me out, or at least put me on the right track. I managed to get through all of them but I'm pretty sure I messed them up.

#1

Define the function f(x) in two pieces by

f(x) = 0 if x ≤ 0, and f(x) = e^(-1/x^2) if x > 0.

Show that the function f(x) is continuous at x = 0. Furthermore, show that f`(0) exists. Your life will be made easier if rather than considering

lim d/dx e^(-1/x^2)

x->0+

you explain why we may instead consider

lim d/dt e^(-t^2)

t-> ∞

Then show that f(ⁿ)(0) exists for all n.

Write the Taylor Series for f(x) centered at 0. Show that this series converges everywhere, but for x > 0 does not converge to f(x).

#2

Suppose that you have the series

∞

∑ an,

n = 1

and assume that

lim |an|^(1/n) = L.

n->∞

Show that if L < 1, then the series converges (absolutely), and if L > 1, the series diverges. Finally, give a specific example of a series where L = 1 and the series converges, and another example where L = 1 and the series diverges.

#3

Assume that the Taylor series for e^z converges for not just for all real numbers z, but for all complex numbers z as well. By setting z = ix, where i^2 = −1 and x is real, show that:

e^ix = cos(x) + i sin(x).

Use this formula to derive Euler’s formula:

e^iπ + 1 = 0.

Thanks in advance.

I apologize that this doesn't look very nice, as well. Not really sure how to do the symbols properly in a text form.