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I'm a UI programmer and I've trying to calculate an effect related to mouse scroll. The behavior I'm noticing is equivalent to

$$x + x^2 + x^3 + x^4 + x^5 + \cdots$$

For the purposes of my app, I only need to go to like $x^3$ before the result is close enough, but I'm wondering if there is a mathematical function or operator, ideally one build into common programming languages, designed for this kind of thing that has greater precision.

Thanks!

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  • $\begingroup$ Welcome to MSE. Please edit and use MathJax to properly format math expressions. $\endgroup$ – Lee David Chung Lin May 11 at 2:01
  • $\begingroup$ What is the typical range of values expected for $x$? $\endgroup$ – Spencer May 11 at 2:11
  • $\begingroup$ @Spencer x should be: $-1 < x < 1$ $\endgroup$ – weotch May 11 at 2:18
  • $\begingroup$ In that case the given answers are sufficient. You should add that detail to your question. $\endgroup$ – Spencer May 11 at 2:19
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I am going to expand on the answer already posted. Feel free to skip the details in my post and read the last line below.

If $x$ is a number between $-1$ and $1$, then you can calculate the infinite sum $x + x^{2} + x^{3} + \dots$.

If we set $$m =x + x^{2} + x^{3} + \dots \,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$ then clearly, multiplying both sides of (1) by $x$, we get $$m\cdot x = x^{2} + x^{3} + \dots.$$ This means we can rewrite the original expression (1) as $m = x + m\cdot x$, and isolating the $m$ gives $m - m\cdot x = x$, i.e., $m(1 - x) = x$, so that we can solve for $m$: $m = \frac{x}{1-x}$.

All of this to say if $x$ satisfies $-1 < x < 1$, then the infinite sum $x + x^{2} + x^{3} + \dots$ can be calculated simply as $\frac{x}{1 - x}$.

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  • $\begingroup$ Holy moly, I hadn't noticed that ... amazing. Thank you! $\endgroup$ – weotch May 11 at 2:21
  • $\begingroup$ @weotch You're welcome! This infinite sum has a special name: geometric series. $\endgroup$ – layman May 11 at 2:22
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It is a geometric series and it can be expressed as it follows (for $|x| < 1)$ \begin{align*} & S(x) = x + x^{2} + x^{3} + \ldots \Rightarrow xS(x) = x^{2} + x^{3} + x^{4} + \ldots \Rightarrow\\\\ & S(x) - xS(x) = S(x)(1 - x) = x \Rightarrow S(x) = \frac{x}{1-x} \end{align*}

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