# Questions tagged [geometric-series]

For questions about or involving geometric series, a series where successive terms have a common ratio.

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### Is the Geometric Series defined at x=0?

The geometric series is usually defined as $\sum_{k=0}^{\infty} a \cdot x^{k}$ where $x$ is on the interval $]-1;1[$, which includes $0$. My Problem is that substituting $x=0$ for the first Term of ...
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### Find the values of a and b from arithmetic and geometric series

The $1^{st}$ , $2^{nd}$ and $3^{rd}$ terms of an arithmetic series are $a, b, a^2$, where $a$ is a negative number. The $1^{st}$, $2^{nd}$ and $3^{rd}$ terms of a geometric series are $a, a^2,b$. Find ...
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### Geometric sequence $a,b,c,d$ and arithmetic sequence $a, \frac{b}{2},\frac{c}{4}, d-70$

The first four terms, given in order, of a geometric sequence $a,b,c,d$ and arithmetic sequence $a, \frac{b}{2},\frac{c}{4}, d-70$, find the common ratio $r$ and the values of each $a,b,c,d$. What I ...
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### How Do I Use Logarithms to calculate the sum of a Geometric Series

I've run into a problem while trying to program something and I'm not much of a mathematician, so I'm hoping someone can jog my memory on how to handle this. I'd like to know how much time it would ...
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### Where have I erred in proving $\sum_{k=0}^\infty ar^k = {a\over 1-r}$?

I'm familiar with the infinite geometric series, its convergence conditions, and the formula for the value of convergence: $$S_{\infty} = {a\over 1-r}\quad |r| < 1$$ ...for an infinite geometric ...
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1 vote
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### Geometric series don’t coincide

Let $j\in \{1,2\}$. For each $j$, assume that $(a^j_n)_{n\ge0}\subset\{0,1\}^\infty$ is a sequence with $a^j_n\in\{1,0\}$ for each $n$. Choose $\xi_1,\xi_2\in (0,\frac{1}{2})$. Let $d^i_n$ be the ...
1 vote
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### How can I evaluate the sum $\sum_{n=1}^{\infty}\frac{(2n)!}{(n!)^2}p^n(1-p)^n$? [duplicate]

How can I evaluate this sum? $$\sum_{n=1}^{\infty}\frac{(2n)!}{(n!)^2}p^n(1-p)^n$$ I know the answer from Wolframalpha, but I'm more curious about how to derive the answer. I was trying to prove that ...
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### Is there a relationship between $\sum _{n=1}^{\infty }\left({\frac {1}{2}}\right)^{n} = 1$ and $\int_{1}^{\infty} \frac{1}{x^2} \,dx = 1$?

A classic example of an infinite series that converges is: ${\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots =\sum _{n=1}^{\infty }\left({\frac {1}{2}}\right)^{n}=1.$...
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### How to calculate an ever increasing number of rounds?

Say you are making a roulette bet and you want to double your bet every time you lose in an attempt to recover what you lost. So, if you lose repeatedly you'd have spent: first round = $10$, second ...
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### Bounding a Series

Prove that $\sum_{i=1}^n \frac{i}{2^i} < 2$ by bounding term-to-term with a geometric series. I thought you'd use $\sum_{i=1}^n (\frac{1}{2})^i$ = 2 somehow but the inequality is not inclusive to ...
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1 vote
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### Is this alternate proof of convergence of geometric series correct?

Show that the sequence {$r^n$} converges to $0$ if $|r|<1$. My attempt Let $a_{n} = r^n$ and $\varepsilon>0$. $|a_{n} - 0| = |r^n| = |r|^n$. Consider $|r|^n < \varepsilon$. Taking log on both ...
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### Average probablity to get a specific outcome on a ten sided dice

A man is playing a game with a ten-sided die, he will roll it every minute and will win if it gives the number $10$. On average, how many minutes would it take him to win the game? My working - ...
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### Find the sum of series $\sum_{k=1}^\infty \frac{1}{k^2+2k}$ [duplicate]

Find the sum of the series.$$\sum_{k=1}^\infty \frac{1}{k^2+2k}$$ Which technique should I use? I tried but I cannot find anything.
1 vote
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### Is there such a thing as a geometric series of a non-constant?

I'm an applied social scientist with an interest in time series analysis. I have a question about the behavior of a 'geometric series' of a non-constant, so to speak. If we had a geometric series like ...
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### Proof of showing difference between reflections with geometric series

I just started studying the field of reflection and Coxeter groups and I'm trying to prove a result but I'm not sure how to do it. I'll sketch the situation first: I'd would like to prove that for ...
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### Evaluating $\sum_{n=1}^\infty\frac{1}{2^{2n-1}}$

What is the summation of this geometric series?$$\sum_{n=1}^\infty\frac{1}{2^{2n-1}}$$ My main confusion is the difference between starting at n=1 versus starting at $n=0$. When you start at $n=1$ I ...