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### Proof of the formula $1+x+x^2+x^3+ \cdots +x^n =\frac{x^{n+1}-1}{x-1}$ [duplicate]

Possible Duplicate: Value of $\sum x^n$ Proof to the formula $$1+x+x^2+x^3+\cdots+x^n = \frac{x^{n+1}-1}{x-1}.$$
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### Easy summation question: $S= 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}\cdots$ [duplicate]

While during physics I encountered a sum I couldn't evaluate: $$S= 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}\cdots$$ Is there a particular formula for this sum and does it converges?
43k views

### Proof of the power series 1 + $x^2$ + $x^3$ + $\ldots$ + $x^n$ = $\frac{1}{1-x}$ [duplicate]

Can anyone show me the proof of this equation: $$\lim_{n \to \infty} 1 + x + x^2 + x^3 + \ldots + x^n = \frac{1}{1-x},$$ where $|x|<1$. Edit: I have then additionally written $x$ on the left ...
11k views

### How to convert a series to an equation? [duplicate]

Possible Duplicate: Value of $\sum\limits_n x^n$ I don't know the technical language for what I'm asking, so the title might be a little misleading, but hopefully I can convey my purpose to you ...
### Why $\sum_{k=0}^{\infty} q^k$ sum is $\frac{1}{1-q}$ when $|q| < 1$ [duplicate]
Why is the infinite sum of $\sum_{k=0}^{\infty} q^k = \frac{1}{1-q}$ when $|q| < 1$ I don't understand how the $\frac{1}{1-q}$ got calculated. I am not a math expert so I am looking for an easy ...