Questions tagged [geometric-series]

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

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How to calculate the annualized geometric return from monthly data point and also have empty months

I have following series of monthly returns with empty months (no investment made). JanX1 --> 0.0125% FebX1 --> 0.009% MarX1 --> 0.024% AprX1 --> 0.047% MayX1 --> 0....
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Forms of Geometric Series

I came across three forms of geometric series and all make sense with the exception of one: $\sum_{n=1}^{\infty}ar^{n+1}$ If the first term of the geometric series has to be a, how does n start at 1 ...
1 vote
1 answer
96 views

Analytic continuation of a lambert series $f(z)=\sum_{n=1}^\infty \frac{2^n z^n}{1-(z/2)^n}$?

Let's define a function for complex $z$ : $$f(z)=\sum_{n=1}^\infty \frac{2^n z^n}{1-(z/2)^n}$$ Lambert series often have an analytic continuation and can even be entire functions so the poles can be ...
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1 vote
2 answers
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Why isn't $1 + x + x^2 \dots = \frac{1}{1-x}$?

In example 8.1 of "Combinatorics and Discrete Mathematics", we consider the generating function: $$ F(x) = \sum_{n=0}^\infty x^n$$ We find that $$(1 + x + x^2 + \dots) - x(1 + x + x^2 + \...
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Stuck while solving a question about geometric sequence

I was trying to solve this question from the book Why math by R.D Driver: Imagine a large piece of paper five-thousandths of an inch thick being torn in half, and the two pieces placed one on top of ...
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2 answers
60 views

summation of $3^k$ [closed]

how do you write the closed form of a sum of the geometric progression of 3^n? Our teacher told us that $2^0+2^1.... 2^n$ is equal to $2^{n+1}-1$ but I am not sure how to apply that to a similar ...
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How to get expressions for interest and principal payments over amortization schedule?

Say I have the following loan: principal: 375000$ yearlyRate = 0.055 (5.5%) years = 30 I already know how to calculate the amortized monthly payment: $$monthlyRate = \dfrac{0.055}{12}$$ $$r = \...
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2 answers
61 views

geometric series with negative exponent

I wonder if it is possible to convert the infinite geometric series with negative exponent to a positive one? Is the calculation is correct ? ∑_(n=1)^∞▒x^(-n) =∑_(n=1)^∞▒〖(〖1/x)〗^n 〗=1/(1-1/x)=x/(x-1)...
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1 answer
60 views

Is this possible to prove statement for any square matrix

I have a statement: $det(I + A + A^2 + ... + A^{2022}) \ge 0$, where A is a square matrix. The task is to prove it and say, is it possible for it to be equal to $0$? I'm preparing for the math contest ...
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2 answers
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Series approximation to $\frac{1}{1-(1+x)^{-n}}$

I'm attempting to find a series approximation to: $$f(x)=\frac{1}{1-(1+x)^{-n}}$$ where $n\in\mathbb{N^+}$ and is a constant, and $x\in\mathbb{R}$ and $0<x<1$. Using Wolfram Alpha, I noted the ...
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3 votes
1 answer
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Closed form of a geometric series without some term

I have to study find the closing form of the following series: \begin{equation} f(1) + f(2) + f(3) + f(5) + f(6) + f(7) + \dots \end{equation} So basically the sum of all terms without the multiple of ...
4 votes
1 answer
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Characteristic function of a random variable with P({X=k}) = $2^{-k}$

Find the characteristic function of a random variable X such that P({X=k}) = $2^{-k}$, $k =1,2,3,4,5, \ldots$ What I was doing is: $$ \phi_x(t) = E(e^{itx}) = \sum^{\infty}_{k=1} 2^{-k} * e^{itk} = \...
1 vote
1 answer
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For which infinite geometric sequences every real number can be represented as the sum of plus-minus of its terms.

For what values of the quotient $q>1$ the following property holds: Every $x\in \mathbb R$ can be represented as $ x = \sum_{n=k}^\infty \pm q^{-n} \tag{*}\label{x} $ for some $k\in \mathbb Z$. ...
1 vote
2 answers
64 views

Every real number can be represented as a sum of plus-minus of the terms of infinite geometric sequence $2^{-n}$.

Every $C\in \mathbb R$ can be represented as $ C = \sum_{n=k}^\infty \pm 2^{-n} $ for some $k\in \mathbb Z$. Trivially, every real number can be represented as $\sum_{n=1}^\infty \pm 2^{-a_n}$ for ...
2 votes
1 answer
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Geometric sum convergence

I am trying to figure out the following problem: Show $\sum_{n=2}^{N} \frac{1}{n^a}≤ \int_1^{N}\frac{1}{x^a} dx$, and use this to prove the convergence of the series for $a>1$. My work: I have $\...
4 votes
2 answers
290 views

Can any $n$ real numbers be part of a geometric sequence?

Ok so, this might be a really silly question, but quite honestly I couldn't find any info about it. So it all begins with a little rather simple question, so I wrote a simple python code, which ...
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-1 votes
1 answer
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Find a coefficient of the geometric series from its sum [closed]

I’m trying to learn about generating functions and need to find the coefficient of $x^{10}$ of this series: $(1+x+x^2+…+x^9)^6.$ I simplified it to $$\frac{(1-x^{10})^6}{(1-x)^6},$$ but don’t know how ...
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Tiling 2 by N rectangle with Tetris shapes

There are similar questions here, but not exactly what I'm looking for. I'm a physics students and we are asked to prove that using the following 6 shapes, The number of ways to build a 2 by N tiling ...
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35 views

Is this generating function correct?

I haven't learned how to calculate generating functions, but I know it has to do with reciprocal functions being equal to regular polynomials, which reminds me of geometric series. I decided to make a ...
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1 vote
1 answer
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How can you check if a series$ ∑_1^{\infty} x^{2}e^{−nx}$ is convergent when $ x \in [0,+∞)$? [closed]

How can you check if a series $$∑_1^{\infty} x^2e^{−nx}$$ is convergent when $x$ belongs to $[0,∞)$ step by step? Can D'Alembert's theorem be used? Could someone write it step by step?
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Find the power-series representation for this function.

Here is a practice exam question we've been given (no answer provided; been scratching my head over it for a while): Here's my work, and answer: Could anyone verify if this is correct? And then, if ...
0 votes
2 answers
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2 cars and the fly problem - calculate the total distance covered by the fly

2 cars approach each other, with 20 km between them. The speed of each car is 10 kmph. At 20 km apart from each, a fly starts traveling from one car towards another at 15 kmph. Once it reaches the ...
-4 votes
1 answer
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Finding minimum of geometric sequence

For the given natural number $0<n\in\mathbb{N}$, define the function as follows: $$f\left(r\right)=2\cdot\sum_{i=0}^{n+2}r^{i-n}+1$$ I want to find the minimum for that function in the domain $r\in\...
0 votes
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Geometric series summation with power which is also geometric.

I was just pondering how will this series converge. Assuming $n \ge 0$, find $$ f(n)=n^{1/2} + n^{1/4} + n^{1/8} + n^{1/16} +\text{...}+ 1 $$ We stop the summation when we hit $1$ i.e. $\lfloor{n^{1/...
2 votes
1 answer
54 views

How to prove the following Dirichlet-series/geometric-series idenity, step by step process?

$$\frac{\zeta(s)}{\zeta(hs)} =\prod_p\left(\frac{1-\frac{1}{p^{hs}}}{1-\frac{1}{p^{s}}}\right) =\prod_p\left(1+\frac{1}{p^s}+\cdots +\frac{1}{p^{(h-1)s}}\right)=\sum_{n\in S_h}\frac{1}{n^s}$$ What is ...
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2 votes
1 answer
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Need help understand some simple algebra

I was looking at some text that involves the following equality, where $z \in \mathbb{C}$ $(1+|z|^2+|z|^4+\dots +|z|^{2n-2})=\frac{|z|^{2n}-1}{|z|^2-1}$ I assumed the text used the formula for ...
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1 answer
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Estimation of a series: why/where is my reasoning wrong'

I have to give an estimation of the value of this series $$\sum_{n = 0}^{+\infty} \dfrac{3^n}{n!}$$ I know this series does converge to $e^3$, but I am not allowed to use this fact. Then here is what ...
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1 vote
1 answer
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Laurent Series of f On Given Annulus

I am attempting to solve the following question and I am encountering some difficulties: Expand the function: $$ f(z) = \frac{z}{z^2 + 2z -3} $$ in powers of z to find a series that is valid for an ...
1 vote
1 answer
71 views

Is it ok for 'r' to be negative in geometric series?

$6+x+y+z+96 ......$ is a geometric series. Here we need to find the value of $x$. Before doing that we need to find the value of $r$. Here, $a=6$, $ar^4=96$ Now, $r^4=16$, then $r=\pm 2$. But, a group ...
2 votes
1 answer
49 views

Geometric series from -i to i

Is it possible to make 2 geometric series from $n = -i, -(i-1), -(i-2),...,(i-1),i$ For a finite series I know that a geometric series looks like that, for $x^n$ $ \sum_0^i = \frac{1-x^{i+1}}{1-x}$
3 votes
2 answers
99 views

limit of a geometric-type series

I came across this series while studying probability theory. Let $$T(x)=\sum_{n=1}^\infty x^n \left(\frac{1}{n}-\frac{1}{n+1}\right),$$ where $T(x)$ converges for some range of $x$. My question is how ...
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0 votes
1 answer
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Simplifying a double sum into a single geometric series

I was looking at how to derive the autocorrelation function of a stationary AR(1) process and ran into the following equality in one derivation I found: $$ \sum_{i=0}^{\infty}\sum_{j=\tau}^{\infty}\...
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1 answer
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Any idea on how WolframAlpha did this sum?

So I've been trying to work out this sum for quite a while: WolframAlpha unfortunately won't supply step by step proofs for this for some reason... As for how i tried to prove this: I looked at sin((...
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How do you compute the result of an equation with discrete values

A bit of background - I had a phonecall appointment with the drs and it wasn't until a few hours after the appointment time that I received the call. I decided to try and model how much extra time I ...
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1 answer
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Probability of printing without a skip.

The old printer in the computer room skips, on average, 1 character in 25 (i.e., the chance of skipping any particular character is 1/25). First part : Determine the probability that a line of 45 ...
3 votes
4 answers
104 views

Finding $S_n$ in terms of n for the sequence (6 + 66 + 666 + 6666 ...)

Sequence given : 6, 66, 666, 6666. Find $S_n$ in terms of n The common ratio of a geometric progression can be solved is $\frac{T_n}{T_{n-1}} = r$, where r is the common ratio and n is the When ...
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1 vote
1 answer
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Infinite sum with exponential

I am solving a question related to Laplace transform and I get the following sum: $$\sum_{n=1}^{\infty}e^{-sn}\left(\frac{1}{1+s}\right)^{n}.$$ The question asks me to show that this is equal to $\...
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2 votes
4 answers
229 views

Tricky limitting sum

Find the sum of the following infinite series, given $|x|<1$ $$2+4x+\frac{9}{2}x^2+\frac{16}{3}x^3+\frac{25}{4}x^4+\frac{36}{5}x^5+\frac{49}{6}x^6+\frac{64}{7}x^7+\frac{81}{8}x^8+ \ldots $$ I have ...
1 vote
1 answer
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Where am I going wrong while trying to figure out the sum of the finite series?

I have to find the Discrete Time Fourier Transform of the function $u[n-a] - u[n-b]$. I can do this in two ways: the first way is to take the DTFT for each function and subtract them: $$\frac {e^{-ajw}...
1 vote
2 answers
90 views

Show that $\frac{1-x^n}{1-x}=0\,$ has a real solution at $x$ equals $-1$

I'm trying to solve this question: $$S_n = 1+x+x^2+x^3...+x^{2021}$$ $$xS_n = x+x^2+x^3+x^4...+x^{2022}$$ Subtracting the bottom equation from the top.. $$S_n(1-x)=1-x^{2022}$$ $$S_n=\dfrac{1-x^{2022}...
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Solving Summations of Two Raised Numbers

I apologize if the title is incorrect, I am not exactly sure how to describe this. I am trying to solve a summation: $$ \sum_{k=0}^n{3^{-k}7^{n-k}} $$ I am not sure how I would go about solving this. ...
6 votes
1 answer
71 views

Can the formula for a series be "reverse engineered" given some of its terms? [duplicate]

Say I have a sequence that goes: 12, 16, 21, 27, 34, 42, 51 and so on in a manner that the increment between $a_1$ and $a_2$ is 4, between $a_2$ and $a_3$ is is 5 etc. The formula which describes this ...
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0 answers
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Is there a formula for finding the index of a binomial tree with this index orientation.

Apologies if this should be in programing. I thought this fit as I'm trying to find a function, not program it, anyway. the binomial tree is structured like this: ...
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1 answer
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Proving an infinite geometric sum

I'm having a bit of trouble determining what I assume to be an infinite geometric sum for the following question in part ii.) I've posted my working below, but I'm not sure how to get the geometric ...
1 vote
1 answer
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If x is some fixed real number, for each $ n \in N $ find the sum of $1-x-x^2-x^3-...-x^n$.

If x is some fixed real number, for each $ n \in N $ find the sum of $1-x-x^2-x^3-...-x^n$. The problem comes from "A friendly introduction to analysis" by Kosmala 2nd ed. My approach to ...
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Partitioning single series into two geometric series to prove convergence

I was asked to determine whether $\frac{1}{3}+ \frac{2}{9} + \frac{1}{27} + \frac{2}{81} + \frac{1}{243} + \frac{2}{729} +...$ converges or diverges and, should it converge, to show what it converges ...
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1 answer
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Convergences of series - Geometric series & quotient rule

I want to check if the following series converge or not. \begin{align*}&1. \ \ \ \sum_{n=0}^{\infty}\frac{4\cdot 4^n+10\cdot 3^n}{5^n} \\ &2. \ \ \ \sum_{n=0}^{\infty}\frac{7^{n+3}}{(n+9)!}\...
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2 votes
0 answers
51 views

What is the sum of geometric series if r is approaching 1 from left side?

We know there is a rule that geometric series converges when $|r|<1$ , and a formula to calculate the sum as $S_n=\frac{a}{1-r}$ or as $S_n=\frac{a(1-r^n)}{1-r}$. My question is: when I take $r$ ...
6 votes
2 answers
93 views

Solving $ \frac{r^{200}-1}{r^{199}-1} = \frac{\alpha}{\beta}$ in a question on Geometric Progressions

Question: Let $a_n$ be the $n^{th}$ term of a geometric progression of positive numbers. Let $$\sum_{n=1}^{200} a_n = \alpha$$ and let $$\sum_{n=1}^{199} a_n = \beta$$ such that $\alpha \neq \beta$. ...
0 votes
1 answer
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How to solve this problem with geometrical series?

I have this question from coursera tutorial. Howver, it has been a long time that I did nothing with maths and cannot solve this problem although I found the formula of geometric series. It is ...
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