If we want to write a function in power series, what's the first n? 0 or 1?

Question

So, for instance $e^x = \sum\limits_{n=0}^\infty \frac{x^n}{n!}$, why do we start from n=0, and not n=1? Why do we care about some zero$^{th}$ term? Do we do the same with other functions? Why?

Answer 1

Suppose

$$f(x)=\sum_{n=0}^\infty a_nx^n,$$

and that this is valid on an interval containing $0$, then

$$f(0)=a_0+\sum_{n=1}^\infty a_n0^n=a_0.$$

So the constant term $a_0$ is the value of the function at $0$. If all of our power series started with $n=1$, say

$$f(x)=\sum_{n=1}^\infty a_nx^n,$$

then $f(0)=0$ all the time.

Answer 2

When referring specifically to taylor series, we have (if $f$ is analytic near $a$) $$ f(x)=\sum_{n=0}^{\infty}f^{(n)}(a)\frac{(x-a)^n}{n!} $$ where the zeroth derivative is $f$ itself ($f^{(0)}:=f$). So in this case, we typically start with $n=0$.

Answer 3

$$\sum\limits_{n=0}^\infty \frac{x^n}{n!}=1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\dots$$ $$\sum\limits_{n=1}^\infty \frac{x^n}{n!}=\,\,\,\,\,\,\,\,\,\,\ x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\dots$$ Which one is equal to the power series of $e^x$ ?