Can someone please give me a graphical interpretation/sense of infinite power series?
Some functions such as exponentials, sines, and cosines are infinite power series, but what does that mean and how does it look like graphically?
I can't really picture what the graph looks like.
The Taylor series (about $x=0$) for $\sin x$ is $$ \sin x=\sum_{n=0}^\infty {(-1)^n x^{2n+1}\over (2n+1)!}. $$ Here's what the sequence of partial sums (blue) $$ S_N(x)=\sum_{n=0}^N {(-1)^n x^{2n+1}\over (2n+1)!} $$ looks like as $N$ varies (here from $0$ to $12$) along with $y=\sin x$ (red):
The idea is as we add more and more terms to the series, it gets closer and closer to $\sin x$ on a longer and longer interval (in this case since the interval of convergence is $(-\infty,\infty)$.)
Here's an example that has a finite interval of convergence to give you an idea what that entails. Using $$ \sum_{n=0}^\infty x^n={1\over 1-x}, \quad -1<x<1, $$ then substituting $-x^2$ in for $x$, we see $$ \sum_{n=0}^\infty (-1)^nx^{2n}={1\over 1+x^2}, \quad -1<x<1. $$ Here, the graph of $y={1\over 1+x^2}$ is shown in red and the $N$th partial sums of the series of various values of $N$ in blue.
Note that as $N$ becomes larger, the blue graph matches the red graph closer and closer on the interval of convergence $-1<x<1$ (highlighted in yellow), but outside of that interval, it does not.
It might be helpful to read this and this which reinforce the idea of a Taylor series as an "infinitely long" Taylor polynomial. Once you are properly grounded in the motivation for and geometric interpretation of Taylor polynomials, you can carry those ideas over to (infinite) Taylor series.
An infinite power series "looks" exactly like a function. Even better, it looks exactly like some function in a given domain!
What I mean by this is, an infinite series such as a Laurent series or a Taylor series, is an approximation to the function within a certain convergence radius. The well known function $e^x$ can be written as a Taylor series who's radius of convergence is infinity. This means that
\begin{gather*} \sum ^\infty_{n=0}\frac{x^n}{n!} =e^x \end{gather*}
For all $x\in \mathbb{R}$. It also happens to be true in the complex plane. There is also a quite nice graphic here which gives you a sense of what the approximation looks like at the $n$th degree.
This isn't the best answer that will show up, but I hope it will solve your immediate questions.
