79 questions linked to/from Value of $\sum\limits_n x^n$
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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?
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### 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 ...
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Possible Duplicate: why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$ If I want to find the formula for $$\sum_{k=1}^n k^2$$ I would do the next: $$(n+1)^2 = a(n+1)^3 + b(n+... 3answers 2k views ### Why  \sum_{k=0}^{\infty} q^k  sum is  \frac{1}{1-q} when |q| < 1 [duplicate] Why is the infinite sum of  \sum_{k=0}^{\infty} q^k = \frac{1}{1-q} when |q| < 1 I don't understand how the \frac{1}{1-q} got calculated. I am not a math expert so I am looking for an easy ... 4answers 15k views ### Proof of the power series 1 + x^2 + x^3 + \ldots + x^n = \frac{1}{1-x} [duplicate] Can anyone show me the proof of why if |x|<1 then:$$ \lim_{n \to \infty} 1+ x^2 + x^3 + \ldots + x^n = \frac{1}{1-x} $$5answers 326 views ### Evaluating the sum of geometric series [duplicate] Possible Duplicate: Value of \sum\limits_n x^n I'm trying to understand how to evaluate the following series:$$ \sum_{n=0}^\infty {\frac{18}{3^n}}. $$I tried following this Wikipedia ... 2answers 2k views ### Simplifying the infinite series \sum_{n = 1}^{\infty} \left(\frac{1}{2}\right)^{3n} [duplicate] Does anyone know of a way to simplify$$ \sum_{n = 1}^{\infty} \left(\frac{1}{2}\right)^{3n} $$to a number? 3answers 236 views ### Prove that 1/2 + 1/4 + 1/8 … = 1 [duplicate] I've often heard that instead of adding up to a little less than one, 1/2 + 1/4 + 1/8... = 1. Is there any way to prove this using equations without using Sigma, or is it just an accepted fact? I need ... 2answers 143 views ### \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac{1}{\infty}=? [duplicate] One of my friends has given me the following problem:$$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac{1}{\infty} = ?$$He said that the answer is 1. He has given his argument as that this is ... 4answers 140 views ### How to calculate k^0+k^1+k^2 + k^3+…+ k^{n-1} [duplicate] How to simplify below expression or convert it to something simpler like k^{n-1}?$$ k^0+k^1+k^2 + k^3+...+ k^{n-1} $$7answers 215 views ### Find the sum of the geometric sequence [duplicate]$$1+ \frac12 + \frac14+\dots+\frac1{2^n}$$To find the sum of the equation you have to find n, the number of terms in the geometric sequence and I don't know how... The answer in the text book is ... 4answers 127 views ### What is the formula of: a^{0} + a^{1} + a^{2} + … + a^{n-1} + a^{n}? [duplicate] What is the formula of:$$a^{0} + a^{1} + a^{2} + ... + a^{n-1} + a^{n} Any ideas?
Possible Duplicate: Value of $\sum\limits_n x^n$ Ive been studying Geometric series and Arithmetic series all day and have struggled to attempt these problems. The Question is to sum up these ...
### Why is $5 + 5z + 5z^2 + … + 5z^{11} = \frac{(5z^{12} - 5)}{(z - 1)}$? [duplicate]
Why is $5 + 5z + 5z^2 + ... + 5z^{11} = \frac{(5z^{12} - 5)}{(z - 1)}$ ? I don't understand how you can rewrite it to that. Z is in this case a complex number: (for example: \$z = 0,8(0,5 + 0,5i\...