I’m in middle school (6th grade) and I have a question related to the exponential function (the teacher couldn’t help me):
We define the exponential as follows:
$$ \exp(t) = \sum_{n = 0}^{\infty} \frac{t^n}{n!}$$
For whole number inputs of $t$, this corresponds to raising $e$ to that power.
I think $\exp(t)$ converges for every possible input of $t$, but I haven’t found a proof yet. If anyone can provide a proof or disproof then that would be helpful. I would also like to know if for some $\exp(t)$ where $t \in U$, $\exp(t)$ will also be $\in U$. However, these are secondary questions and you can ignore them (but do answer if you want).
We also define $\exp(t)$ for complex numbers, and matrices.
Today I thought about the exponential of a function:
$\exp(f(t))$.
For example, let’s take $\exp(t^2)$. This would result in something like:
$\exp(t^2) = \sum_{n = 0}^{\infty} \frac{(t^2)^n}{n!} = 1 + t^2 + \frac{t^4}{2!} + \frac{t^6}{3!} + \cdots$
Which we could write as a regular polynomial as
$1 + t^2 + \frac{1}{2}t^4 + \frac{1}{6}t^6 + \frac{1}{24}t^8 + \cdots$
Another example: $\exp\left(\sin(t)\right)$:
$1 + \sin(t) + \frac{1}{2}\sin^2 (t) + \frac{1}{6}\sin^3 (t) + \cdots$
Actually, this is the same thing as raising $e$ to the power of a function. If we took $\exp(e^x)$ it would be the same as $e^{(e^x)}$:
$1 + \frac{1}{2}e^x + \frac{1}{6}e^{2x} + \cdots$
Question: Does the exponential “converge” for function inputs? I mean it certainly approaches one single infinite series (polynomial for polynomial inputs) but do we consider that convergence? Also, is there anything useful you can take away from taking the exponential of a function?
