Question

Taylor/Maclaurin Series help please?

I understand how to construct a Taylor/Maclaurin series. However, my teacher stated that, for example, we can find the power series for \(e^{x^2}\) by plugging in \(x^2\) for x in the power series for \(e^x\).
When I asked him how we could simply "substitute," he said that he has never found a satisfactory proof. Does anyone know of a proof of this composite function property, or tell me how to prove it?

Answer 1

a variable is just a variable

Answer 2

there of course could be a concern about convergence. but not in the case of \(e^x\) since it converges for all \(x\)

Answer 3

including \(x=x^2\)

Answer 4

example
\[\large \frac{1}{1-\heartsuit}=\sum\heartsuit^n\] i can put anything inside \(\heartsuit\) i chose, and it will converge so long as \(|\heartsuit|<1\)

Answer 5

the same reason when you have y = f(x), you can replace x with other value say t, i.e y = f(t), as long as t is in the domain of f.

Every series has interval of convergent. Meaning, the value of the function can be represented by the series if the thing you plug in is in that interval.

Using the example Satellite73 mention,
1/(1-x) = sum{n=0 to inf} x^n, as long as |x| < 1.

if i have 1/(1-sinx), I can do
1/(1-x) = sum{n=0 to inf) (sinx)^n, as long as |sinx| < 1

Answer 6

the interval of convergence of the series that represents e^x is (-inf,inf).
x^2 is also in (-inf,inf) no matter what x is. That's why you're allowed to do substitution.