<s>[INST] What is the closed form of infinite sum $x+x^4+x^7+x^{10}+...$? I know that $1+x+x^2+x^3+...=\frac{1}{1-x}$, $1+x^2+x^4+...=\frac{1}{1-x^2}$. It seems to follow some rules of geometric series. But I'm confused how to get the title from here. Thanks
 [/INST] It's always nice, if we want to see if the sum is geometric, to do factoring to ensure that the first term of the sum is $1$, so factor out an $x$:
$$x + x^4 + x^7 + x^{10} + ... = x(1 + x^3 + x^6 + x^9 + ...)$$
The parenthetical expression is a geometric series in which the ratio is $x^3$. Then as a result, by the formula for an infinite geometric sum,
$$1 + x^3 + x^6 + x^9 + ... = \frac{1}{1-x^3}$$
Thus, bringing in the factored-out $x$,
$$x + x^4 + x^7 + x^{10} + ... = \frac{x}{1-x^3}$$
Note that the formula for a geometric sum only holds when the ratio is less than $1$ in magnitude. Thus we require $|x^3| < 1$. (This can be simplified to $|x| < 1$ as $|x^n|=|x|^n$ and then you just take the cube root.)
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