Questions tagged [geometric-series]

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

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1
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2answers
38 views

Exponentially weighted average, sum of weights

I'm just wondering how to actually prove that the weights of an exponentially weighted average sum to 1, like $(1-\alpha)^n + \displaystyle\sum_{i=1}^n \alpha (1-\alpha)^{n-i} = 1$ where $\alpha \in (...
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3answers
54 views

Why $\sum_{i=1}^{k} \frac{1}{p^i} = \frac{p^{-k}-1}{1-p}$?

I have the statement: $$\sum_{i=1}^{k} \frac{1}{p^i} = \frac{p^{-k}-1}{1-p}$$ without a clear step explanation why. I've tried expanding the LHS and rewriting the RHS, but I don't see it yet. $$\sum_{...
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1answer
32 views

Is it valid to use the Geometric Series Test for Power Series?

My textbook (Early Transcendentals 8th e., James Stewart) advises that in general to find the interval of convergence of a power series we should use the Ratio or Root Tests. However, I found that the ...
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2answers
43 views

How do you use generating functions in this problem?

Let's say that I would like to find all the ways I could make a dollar using pennies, nickels, dimes, quarters, half-dollars, and dollar coins, assuming that all coins of a type are indistinguishable. ...
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1answer
45 views

Stationary distribution for Amount of Cash in an ATM

I'm a little stuck on how to solve this problem. Suppose I have an ATM that starts with zero dollars. Every morning, a truck will deposit a random amount of dollar bills into the ATM so people can ...
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1answer
49 views

Finding Big Theta of Recurrence Relation $8T(n/3)+n^2/\log_3(n)$

Getting stuck on finding the geometric series after finding the sum for this recurrence. I didn't think master theorem applied to this because I thought n^2/log3(n) was not a polynomial but I could be ...
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1answer
15 views

How do you apply finite geometric series in order to determine the distance a bouncing ball travels up and down at the 10th bounce?

My question is very similar to this one, except I would like to determine the distance traveled immediately after the 10th bounce. Assume the ball is let go from 1 meter above the ground and each ...
1
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1answer
35 views

Sum of squares of natural numbers and geometric series

$$S_n = 0\cdot4^0 + 1^2\cdot4^1 + 2^2\cdot4^2 + 3^2\cdot4^3 + \ldots + n^2\cdot4^n$$ ie- $S_n = \sum\limits_{i=0}^ni^2\cdot4^i$. I need help with this problem. I integrated it using ILATE rule and ...
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0answers
21 views

Simplifying a geometric series with three distinct terms

The question is really quite simple - I would like to simplify the following series: $\sum_{z=1}^{\infty}\sum_{y=1}^{\infty}\left(\frac{1}{2}\right)^y(1-e^{-y})^2e^{-y(x+z-2)}$ My approach: It is ...
2
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1answer
63 views

Is the answer to this summation correct?

Consider the summation below: $$\sum_{k=1}^\infty(2k+1)x^{2k}$$ The problem asks to find what the mentioned summation is equal to. The solution provided in the book starts like the following: $$x^3+x^...
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1answer
44 views

Sequence/series with triangular(?) pattern emerging

What i thought was a geometric series of the following form: $$N = \sum_{i=1}^n[N_{i-1} + (T-N_{i-1})P]$$ where $$ lim_{i \to \infty} N = T$$ and $$N_{i=0}=0$$ I find the series to do the following: $$...
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1answer
34 views

How do I find the infinite sum of z/(z-2)

To find the infinite sums, in general, the following approach seems to work: $\sum_{0}^{\infty}z^{n+1} = z/(1-z)$, where $|z|<1$. This is just different version from the standard equation for power ...
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2answers
129 views

How to integrate $\int_0^{\infty} \frac{x}{e^x+1} dx$

I want to integrate it explicitly. I looked up if this had an exact solution, and I got that it is $\frac{\pi^2}{12}$ here: https://www.integral-calculator.com/#expr=x%2F%28e%5Ex%2B1%29&lbound=0&...
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0answers
14 views

Convex inequality in Arithematic geometric mean

$|I|^{1/2} + |J|^{1/2}\leq\ [ 2(|I|+|J|) ]^{1/2}$. Generalise this inequality for $n$, where $n$ is the power, which is $1/2$ here in LHS?
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2answers
47 views

Partial sum of Geometric sequence [duplicate]

Can someone explain to me how the following summation goes from the left to the right: $\sum_{i=1}^{r}n(1-p)^{i-1} = \frac{n(1-(1-p)^r)}{p}$. I have used the formula for a Geometric series. My common ...
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1answer
36 views

Alternating geometric series, not sure what i am looking at

I have written out a series describing a model system but i cannot find if there is a historical representation of the like series: $$N_s= N\sum_{i=1}^n(-1)^{i-1}P^i$$ It appears to be an alternating ...
0
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1answer
27 views

Adding decreasing power of the same number

Simpler ways of writing for example $a^{n}$ + $a^{n-1}$ + $a^ {n-2}$ + ... + a= what is this even called ?
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1answer
16 views

Find $a,b,c$ of a progression

The numbers $a,b,c$ are in an arithmetic progression with $a+b+c=124$ If $a,b,c$ are the third, thirteenth and fifteenth terms of an arithmetic progression, find $a,b,c$ There are no hints in the ...
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2answers
71 views

Geometric sum with binomial coefficient

I'm looking for any kind of formula or insight concerning $$ f(k,N)=\sum_{n=0}^N\binom{n}{k}\lambda^n $$ There are loads of binomial identities, so maybe I was just missing something, but I couldn't ...
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1answer
56 views

Problem with infinite series of residues

I tried solving the integral $$ I := \int_{-\infty}^{\infty} \frac{x \exp(\mu x)}{\exp(\nu x)-1}\,dx,~~\text{where}~~\text{Re}(\nu)>\text{Re}(\mu) >0$$ using the calculus of residues. I found ...
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2answers
50 views

Find the first term in a geometric progression

The exercise reads as follows: The sum of the first 5 terms in a geometric progression is 62. The 5th, 8th and 11th term of this geometric sequence are also the 1st, 2nd and 10th term of an ...
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2answers
47 views

Find the numbers $a,b,c,d$ in a geometric sequence, knowing that $a+1$, $b+6$, $c+6$, $d-4$ are in an arithmetic sequence

The exercise reads as follows: Find the numbers $a,b,c,d$ in a geometric sequence, knowing that $a+1$, $b+6$, $c+6$, $d-4$ are in an arithmetic sequence. I am interested in finding out the steps I ...
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1answer
39 views

Is ${s_n} = \sum\limits_{i = 1}^n {\dfrac{1}{{{3^{i - 1}}}}} = \dfrac{3}{2}\left( {1 - \dfrac{1}{{{3^n}}}} \right)$?

Here in example 4 this result is shown. ${s_n} = \sum\limits_{i = 1}^n {\dfrac{1}{{{3^{i - 1}}}}} = \dfrac{3}{2}\left( {1 - \dfrac{1}{{{3^n}}}} \right)$ Now without looking at its solution I was ...
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5answers
59 views

Find variables from a multiplication of geometric sequences

Given $$f(x) = \sum_{i=0}^\infty a_ix^i$$ And $$f(x)(1+2x+2x^2+x^3) = \frac{1}{(1-x)^3}$$ Find the values of $a_0$, $a_1$, and $a_2$. I try to expand them but it doesn't seem to lead me to any ...
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1answer
35 views

Distribution of range of N geometric random variables

Suppose I have N i.i.d geometric random variables with parameter $p$: $X_1, X_2, \dots, X_N$. I want to know how large the range can often be. I.e. For a fixed significance level $\alpha$, find $\...
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3answers
85 views

Triple summation verification $\sum_{1\le i\lt j\lt k}\frac{1}{2^i3^j5^k}$

I am attempting the following triple summation. It would be great if someone would verify whether what I've done is correct. $$\sum_{1\le i\lt j\lt k}\frac{1}{2^i3^j5^k}$$ $$\begin{aligned}\sum_{1\...
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1answer
45 views

Roots of a quartic form a geometric progression [closed]

Determine all real values of the parameter $a$ for which the equation $$16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0$$has exactly four distinct real roots that form a geometric progression.
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2answers
39 views

The series $1 + v^2 + v^4 + v^6 + v^8 + …$ is gamma squared

Can anyone tell me anything about the series $ 1 + v^2 + v^4 + v^6 + v^8 ... $ Does it have a name? Does it have any special properties? I ask because $ \gamma = \sqrt{1 + v^2 + v^4 + v^6 + v^8 ...} $ ...
1
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1answer
105 views

Tricky mortgage rate question

Can anybody suggest me how I might solve the following equation? $$(1+x)^{300} -125x(1+x)^{300} = 1$$ where x is the unknown that I want to solve for. The actual question in full is Q. Assume that ...
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3answers
64 views

Mistake converting $2.3151515…$ to a fraction?

So I think I made a mistake but I just want to confirm I'm getting this right: In the example, we had the number $2.3151515...$ and were supposed to convert this number into a fraction. It was then ...
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0answers
8 views

Formalization of simple case of rejection sampling

I am watching online lectures on rejection sampling, and I would like to play with it a little. For this, I am trying to do a very simple exercise proposed in this nice video. Let's suppose that we ...
2
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2answers
88 views

Basic Geometric Series Question-Stuck

I'm studying Calc 2 and I have a basic series question. A geometric series $\sum _{n=0}^{\infty }\:Ar^n$ is convergent if |r|<1 and the sum equals $\frac{a}{1-r}$ if the series is convergent. ...
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1answer
34 views

Evaluating $\sum_{k=0}^{m-1}\left(e^{-2\pi in/m}\right)^k$

Let $k,m,n \in \mathbb{Z}$ and consider $\sum_{k=0}^{m-1}\left(e^{-2\pi in/m}\right)^k$. I was wondering why $$ \sum_{k=0}^{m-1}\left(e^{-2\pi in/m}\right)^k = \begin{cases} m , \ \ \ \ \ \ n/m \in \...
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2answers
146 views

Solving finite and infinite nested square roots of 2 - yet another interesting approach

Consider the following consecutive equalities: $\sqrt2=2\cos(\frac{1}{4})\pi$ $\sqrt{2-\sqrt2}=2\cos(\frac{3}{8})\pi$ $\sqrt{2-\sqrt{2-\sqrt{2}}}=2\cos(\frac{5}{16})\pi$ $\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{...
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1answer
22 views

Is it possible to combine an arithmetic an geometric sequence, but first adding and then multiplying?

picture of solution for when you multiply first and add after link of original discussion of the picture: Combined geometric and arithmetic series partial sum In this picture, you can see a way of ...
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1answer
44 views

Find $\mathbb{E}[X(X-1)]$ for the geometric distribution without using derivation

I'm trying to find the $\mathbb{E}[X(X+1)]$ for the geometric distribution. Everywhere I've looked explains how to do it using the derivative but I have not been taught that method. Is there another ...
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2answers
40 views

Closed-form solution to an infinite series?

I have the following series that I've confirmed on matlab to have some closed-form solution, but I can't find it through trial-and-error, and I definitely don't have the math background to just solve ...
0
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1answer
27 views

Finding the Sum of a Geometric Series in Finance

I have been stuck on these two problems forever, I have tried using the Sum of a Finite Geometric Series formula, but I cannot get to the intended final product of 200(1.005^(n)-1). I even double ...
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1answer
40 views

Convergence of a geometric series but the sum takes square integers

I want to know if there is a closed form for the convergence of a series of this kind: $$\sum_{j=1}^\infty r^{j(j-1)/2}$$ given that $0<r<1$. As it is possible to see the terms of the sum are $...
0
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1answer
28 views

Unsure of steps of how to solve infinite summation

I am trying to find the convergent value for the following summation: $$\sum_{a=0}^∞ a \left( \frac{x-1}{x}\right)^{\!a}$$ This sum seems to be convergent by ratio test as $$\frac{x-1}{x} < 1$$ but ...
0
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2answers
33 views

How do you find the common ratio of a geometric sequence if not given the first term?

The only given values are the sum of an infinite geometric series which is equal to 9/2, and the second term which is equal to -2. How do I find the common ratio here?
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1answer
80 views

Is there a deeper connection among $\int_0^1 x^{e-2} dx$, $\int_0^{\infty}\{x\}e^{-x}\,dx$, and $\sum_{n=1}^{\infty} e^{-n}$?

Consider the following integrals $I_1,I_2$: $$ I_1=\int _1^{\infty} \lfloor x\rfloor e^{-x}\,dx = \sum_{n=1}^{\infty}n \int_{n}^{n+1}e^{-x}\,dx = (e-1)\sum_{n=1}^{\infty} n e^{-n-1} = \frac{e-1}{(e-1)^...
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1answer
17 views

Determining the change in $P(t)$ over the infinitesimal time $dt$

A certain security has a price given by the following stochastic process: $$P(t) = S(t)e^{(r-q)r}, \hspace{20 pt} 0\le t \le T, \tau = T - t$$ where $S(t)$ is the price of a security following ...
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0answers
21 views

Geometric distribution question with a small show that section

If a football team has to leave a tournament after they lose. & in each match they have a probability $q$ of losing. Assuming each match is independent. IF they play X total matches. Am I right in ...
2
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1answer
62 views

Closed form of geometric series $\sum_{i=1}^n p^{i+1}$

I know that $\sum_{i=1}^n p^i = \frac{p-p^{n+1}}{1-p}$, but I am not sure how the i+1 factors into the closed form for $\sum_{i=1}^n p^{i+1}$, what is the closed form for the second sum?
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2answers
59 views

How to find the correct formula to calculate the sum of a geometric series.

I have a formula for a series which looks like this: $\sum_{i=2}^{8}(-2)^i$ And I see that this series gives me the following output: 4 - 8 + 16 - 32 + 64 - 128 + 256 Which gives a sum of 172 - ...
3
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1answer
91 views

Geometric sum converges to 1/2

$$\sum_{n=0}^\infty\frac{1}{3^{2^n}-3^{-2^n}}=\frac{1}{2}$$ I assumed this is a geometric series, but im not 100% sure about it. I checked if its convergent anyway, and it is, but other than that I ...
0
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0answers
14 views

Sum of Geometrical Matrix Series with Variance Notation

I am looking for a formula which has the following structure: $$\mathbf{S_n}=\mathbf{A}+\mathbf{BA}\mathbf{B}^\prime+\mathbf{B}^\mathbf{2}\mathbf{A}\mathbf{B}^{\mathbf{2}^\prime}+\mathbf{B}^\mathbf{3}\...
3
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0answers
71 views

Can the Bernoulli numbers be viewed as a 'renormalization' of a finite geometric series with term $e^{-x}$, by integrating over $(-1,0)$?

I was playing around with ways to calculate Bernoulli numbers; for this post I will take their generating function as $x/(1-e^{-x})$, that is, $$ \sum_{n=0}^{\infty} \frac{B_n}{n!}x^n = \frac{x}{1-e^{-...

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