Questions tagged [geometric-series]
For questions about or involving geometric series, a series where successive terms have a common ratio.
648
questions
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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 (...
1
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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.
$$\sum_{...
1
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1answer
32 views
Is it valid to use the Geometric Series Test for Power Series?
My textbook (Early Transcendentals 8th e., James Stewart) advises that in general to find the interval of convergence of a power series we should use the Ratio or Root Tests. However, I found that the ...
1
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2answers
43 views
How do you use generating functions in this problem?
Let's say that I would like to find all the ways I could make a dollar using pennies, nickels, dimes, quarters, half-dollars, and dollar coins, assuming that all coins of a type are indistinguishable. ...
3
votes
1answer
45 views
Stationary distribution for Amount of Cash in an ATM
I'm a little stuck on how to solve this problem. Suppose I have an ATM that starts with zero dollars. Every morning, a truck will deposit a random amount of dollar bills into the ATM so people can ...
1
vote
1answer
49 views
Finding Big Theta of Recurrence Relation $8T(n/3)+n^2/\log_3(n)$
Getting stuck on finding the geometric series after finding the sum for this recurrence. I didn't think master theorem applied to this because I thought n^2/log3(n) was not a polynomial but I could be ...
0
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1answer
15 views
How do you apply finite geometric series in order to determine the distance a bouncing ball travels up and down at the 10th bounce?
My question is very similar to this one, except I would like to determine the distance traveled immediately after the 10th bounce.
Assume the ball is let go from 1 meter above the ground and each ...
1
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1answer
35 views
Sum of squares of natural numbers and geometric series
$$S_n = 0\cdot4^0 + 1^2\cdot4^1 + 2^2\cdot4^2 + 3^2\cdot4^3 + \ldots + n^2\cdot4^n$$
ie- $S_n = \sum\limits_{i=0}^ni^2\cdot4^i$.
I need help with this problem. I integrated it using ILATE rule and ...
2
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0answers
21 views
Simplifying a geometric series with three distinct terms
The question is really quite simple - I would like to simplify the following series:
$\sum_{z=1}^{\infty}\sum_{y=1}^{\infty}\left(\frac{1}{2}\right)^y(1-e^{-y})^2e^{-y(x+z-2)}$
My approach: It is ...
2
votes
1answer
63 views
Is the answer to this summation correct?
Consider the summation below:
$$\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^...
0
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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:
$$...
1
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1answer
34 views
How do I find the infinite sum of z/(z-2)
To find the infinite sums, in general, the following approach seems to work:
$\sum_{0}^{\infty}z^{n+1} = z/(1-z)$, where $|z|<1$. This is just different version from the standard equation for power ...
5
votes
2answers
129 views
How to integrate $\int_0^{\infty} \frac{x}{e^x+1} dx$
I want to integrate it explicitly. I looked up if this had an exact solution, and I got that it is $\frac{\pi^2}{12}$ here: https://www.integral-calculator.com/#expr=x%2F%28e%5Ex%2B1%29&lbound=0&...
0
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0answers
14 views
Convex inequality in Arithematic geometric mean
$|I|^{1/2} + |J|^{1/2}\leq\ [ 2(|I|+|J|) ]^{1/2}$. Generalise this inequality for $n$, where $n$ is the power, which is $1/2$ here in LHS?
1
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2answers
47 views
Partial sum of Geometric sequence [duplicate]
Can someone explain to me how the following summation goes from the left to the right:
$\sum_{i=1}^{r}n(1-p)^{i-1} = \frac{n(1-(1-p)^r)}{p}$.
I have used the formula for a Geometric series. My common ...
0
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1answer
36 views
Alternating geometric series, not sure what i am looking at
I have written out a series describing a model system but i cannot find if there is a historical representation of the like series:
$$N_s= N\sum_{i=1}^n(-1)^{i-1}P^i$$
It appears to be an alternating ...
0
votes
1answer
27 views
Adding decreasing power of the same number
Simpler ways of writing for example $a^{n}$ + $a^{n-1}$ + $a^ {n-2}$ + ... + a=
what is this even called ?
0
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1answer
16 views
Find $a,b,c$ of a progression
The numbers $a,b,c$ are in an arithmetic progression with $a+b+c=124$
If $a,b,c$ are the third, thirteenth and fifteenth terms of an arithmetic progression, find $a,b,c$
There are no hints in the ...
1
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2answers
71 views
Geometric sum with binomial coefficient
I'm looking for any kind of formula or insight concerning
$$
f(k,N)=\sum_{n=0}^N\binom{n}{k}\lambda^n
$$
There are loads of binomial identities, so maybe I was just missing something, but I couldn't ...
1
vote
1answer
56 views
Problem with infinite series of residues
I tried solving the integral
$$ I := \int_{-\infty}^{\infty} \frac{x \exp(\mu x)}{\exp(\nu x)-1}\,dx,~~\text{where}~~\text{Re}(\nu)>\text{Re}(\mu) >0$$
using the calculus of residues. I found ...
0
votes
2answers
50 views
Find the first term in a geometric progression
The exercise reads as follows:
The sum of the first 5 terms in a geometric progression is 62. The 5th, 8th and 11th term of this geometric sequence are also the 1st, 2nd and 10th term of an ...
0
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2answers
47 views
Find the numbers $a,b,c,d$ in a geometric sequence, knowing that $a+1$, $b+6$, $c+6$, $d-4$ are in an arithmetic sequence
The exercise reads as follows:
Find the numbers $a,b,c,d$ in a geometric sequence, knowing that $a+1$, $b+6$, $c+6$, $d-4$ are in an arithmetic sequence.
I am interested in finding out the steps I ...
0
votes
1answer
39 views
Is ${s_n} = \sum\limits_{i = 1}^n {\dfrac{1}{{{3^{i - 1}}}}} = \dfrac{3}{2}\left( {1 - \dfrac{1}{{{3^n}}}} \right)$?
Here in example 4 this result is shown.
${s_n} = \sum\limits_{i = 1}^n {\dfrac{1}{{{3^{i - 1}}}}} = \dfrac{3}{2}\left( {1 - \dfrac{1}{{{3^n}}}} \right)$
Now without looking at its solution I was ...
2
votes
5answers
59 views
Find variables from a multiplication of geometric sequences
Given $$f(x) = \sum_{i=0}^\infty a_ix^i$$
And $$f(x)(1+2x+2x^2+x^3) = \frac{1}{(1-x)^3}$$
Find the values of $a_0$, $a_1$, and $a_2$.
I try to expand them but it doesn't seem to lead me to any ...
2
votes
1answer
35 views
Distribution of range of N geometric random variables
Suppose I have N i.i.d geometric random variables with parameter $p$: $X_1, X_2, \dots, X_N$. I want to know how large the range can often be. I.e. For a fixed significance level $\alpha$, find $\...
6
votes
3answers
85 views
Triple summation verification $\sum_{1\le i\lt j\lt k}\frac{1}{2^i3^j5^k}$
I am attempting the following triple summation. It would be great if someone would verify whether what I've done is correct.
$$\sum_{1\le i\lt j\lt k}\frac{1}{2^i3^j5^k}$$
$$\begin{aligned}\sum_{1\...
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.
0
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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 ...
1
vote
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 ...
0
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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
votes
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.
...
0
votes
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 \...
4
votes
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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votes
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 ...
1
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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 ...
0
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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 ...
0
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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}$$ given that $0<r<1$. As it is possible to see the terms of the sum are $...
0
votes
1answer
28 views
Unsure of steps of how to solve infinite summation
I am trying to find the convergent value for the following summation:
$$\sum_{a=0}^ā a \left( \frac{x-1}{x}\right)^{\!a}$$
This sum seems to be convergent by ratio test as $$\frac{x-1}{x} < 1$$ but ...
0
votes
2answers
33 views
How do you find the common ratio of a geometric sequence if not given the first term?
The only given values are the sum of an infinite geometric series which is equal to 9/2, and the second term which is equal to -2. How do I find the common ratio here?
1
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1answer
80 views
Is there a deeper connection among $\int_0^1 x^{e-2} dx$, $\int_0^{\infty}\{x\}e^{-x}\,dx$, and $\sum_{n=1}^{\infty} e^{-n}$?
Consider the following integrals $I_1,I_2$:
$$
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)^...
2
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1answer
61 views
0
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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 ...
1
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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
votes
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?
1
vote
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
votes
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
votes
0answers
14 views
Sum of Geometrical Matrix Series with Variance Notation
I am looking for a formula which has the following structure:
$$\mathbf{S_n}=\mathbf{A}+\mathbf{BA}\mathbf{B}^\prime+\mathbf{B}^\mathbf{2}\mathbf{A}\mathbf{B}^{\mathbf{2}^\prime}+\mathbf{B}^\mathbf{3}\...
3
votes
0answers
71 views
Can the Bernoulli numbers be viewed as a 'renormalization' of a finite geometric series with term $e^{-x}$, by integrating over $(-1,0)$?
I was playing around with ways to calculate Bernoulli numbers; for this post I will take their generating function as $x/(1-e^{-x})$, that is,
$$
\sum_{n=0}^{\infty} \frac{B_n}{n!}x^n = \frac{x}{1-e^{-...
