TITLE: Define $1+x+x^2+\cdots+x^n$ recursively
QUESTION [5 upvotes]: Let $x \neq 1$ be a real number. Define the following sum recursively:
$$
1+x+x^2+\cdots+x^n
$$
My attempt:
Using summation notation we can write the sum as $\sum_{i=0}^{n} x^i$. Now define
$$
\sum_{i=0}^{0}x^i:=x^0=1 \quad \text{and} \quad \sum_{i=0}^{n}x^i:=\sum_{i=0}^{n-1}x^i+x^n
$$
It seems to me correct because if we use the definition for $n=3$, then
$$
\sum_{i=0}^{3}x^i=\sum_{i=0}^{2}x^i+x^3=(\sum_{i=0}^{1}x^i+x^2)+x^3=((\sum_{i=0}^{0}x^i+x)+x^2)+x^3=1+x+x^2+x^3
$$
In other words, we reach the base case. However, I'm unsure if this is correct. I might have misunderstood what a recursive definition is.

REPLY [3 votes]: $$f_n(x) = 1 + x + \ldots + x^n.$$
It can be rewriten as
$$f_n(x) = 1 + x\cdot\left(1 + \ldots + x^{n-1}\right),$$
or
$$f_n(x) = 1 + x\cdot f_{n-1}(x)$$
for $n>0$ and
$$f_0(x) = 1.$$