Series

Abstract

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• Also known as Geometric Series
• Sum of numbers in a sequence

Finite Series Partial Sum

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$$S_n=\frac{a(1-r^n)}{1-r}$$

What is the proof?

Given sequence $S_n=a+ar+a{r}^2+\cdots +a{r}^{n-1}$. We can multiply both sides with a common ratio $r$, and we get $r{S}_n=ar+a{r}^2+a{r}^3+\cdots +a{r}^n$.

We then subtract the second equation from the first, so most of the terms cancel out, $S_n-r{S}_n=a-a{r}^n$.

To solve for $S_n$, we perform factorisation $S_n(1-r)=a(1-r^n)$, and we get $S_n=\frac{a(1-r^n)}{1-r}$.

References

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• 【漫士科普】为什么说数学思维就是抽象 - YouTube