name of theorem for infinite order polynomial limit for small x

Question

is there a name for this property?

$\mathop{\operatorname{lim}}_{x\to 0} \sum_{n=1}^\infty x^n = \frac{x}{1-x}$

I have seen it in a derivation but want to know where it comes from.

Answer 1

Firstly, the formula as written is false. On the left hand side you take a limit as $x$ approaches $0$, yet on the right hand side you have a function of $x$. The correct formula is that for all $x\in \mathbb R$ with $|x|<1$ it holds that $$\sum_{n=1}^\infty x^n=\frac{x}{1-x}$$

which is known as the sum of a geometric series (it actually holds true for all complex numbers $z$ with $|z|<1$ and the formula fails for any number with absolute value bigger than or equal to $1$). Here is an informal argument (that can be turned completely formal) for proving it: believing (this is the informal part) that the series actually converges, say to $L$, we have that (with a bit more formal stuff lurking around) $$(1-x)L=(1-x)\cdot \sum_{n=1}^\infty x^n=\frac{x}{1-x}=\sum_{n=1}^\infty x^n - \sum_{n=1}^\infty x^{n+1}=x$$ from which the formula follows. There are other ways of showing the formula holds, and, of course, they all require some amount of limit formalism.

One says that the function $\frac{x}{1-x}$ admits a power series representation with center $x=0$ and radius of convergence $1$.

Answer 2

You can show that

$$\sum_{n=1}^{N-1}x^n=\frac{x-x^N}{1-x}$$

For $|x|<1$ you can take the limit $N\rightarrow \infty$ and you obtain the result. It's called geometric series.