Taylor Series/Limits/Convergence Test
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.
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)
you explain why we may instead consider
lim d/dt e^(-t^2)
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).
Suppose that you have the series
n = 1
and assume that
lim |an|^(1/n) = L.
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.
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.