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....
0
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0
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19
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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
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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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2
answers
71
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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
$$(1 + x + x^2 + \dots) - x(1 + x + x^2 + \...
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0
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0
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33
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Stuck while solving a question about geometric sequence
I was trying to solve this question from the book Why math by R.D Driver:
Imagine a large piece of paper five-thousandths of an inch thick being torn in half, and the two pieces placed one on top of ...
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0
votes
2
answers
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summation of $3^k$ [closed]
how do you write the closed form of a sum of the geometric progression of 3^n? Our teacher told us that $2^0+2^1.... 2^n$ is equal to $2^{n+1}-1$ but I am not sure how to apply that to a similar ...
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How to get expressions for interest and principal payments over amortization schedule?
Say I have the following loan:
principal: 375000$
yearlyRate = 0.055 (5.5%)
years = 30
I already know how to calculate the amortized monthly payment:
$$monthlyRate = \dfrac{0.055}{12}$$
$$r = \...
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votes
2
answers
61
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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)...
0
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1
answer
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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
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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
61
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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:
$$
\phi_x(t) = E(e^{itx}) = \sum^{\infty}_{k=1} 2^{-k} * e^{itk} = \...
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1
answer
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For which infinite geometric sequences every real number can be represented as the sum of plus-minus of its terms.
For what values of the quotient $q>1$ the following property holds:
Every $x\in \mathbb R$ can be represented as
$
x = \sum_{n=k}^\infty \pm q^{-n} \tag{*}\label{x}
$
for some $k\in \mathbb Z$.
...
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1
vote
2
answers
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Every real number can be represented as a sum of plus-minus of the terms of infinite geometric sequence $2^{-n}$.
Every $C\in \mathbb R$ can be represented as
$
C = \sum_{n=k}^\infty \pm 2^{-n}
$
for some $k\in \mathbb Z$.
Trivially, every real number can be represented as $\sum_{n=1}^\infty \pm 2^{-a_n}$ for ...
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2
votes
1
answer
27
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Geometric sum convergence
I am trying to figure out the following problem: Show $\sum_{n=2}^{N} \frac{1}{n^a}≤ \int_1^{N}\frac{1}{x^a} dx$, and use this to prove the convergence of the series for $a>1$.
My work: I have $\...
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2
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Can any $n$ real numbers be part of a geometric sequence?
Ok so, this might be a really silly question, but quite honestly I couldn't find any info about it. So it all begins with a little rather simple question, so I wrote a simple python code, which ...
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1
answer
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Find a coefficient of the geometric series from its sum [closed]
I’m trying to learn about generating functions and need to find the coefficient of $x^{10}$ of this series: $(1+x+x^2+…+x^9)^6.$ I simplified it to $$\frac{(1-x^{10})^6}{(1-x)^6},$$ but don’t know how ...
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0
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Tiling 2 by N rectangle with Tetris shapes
There are similar questions here, but not exactly what I'm looking for. I'm a physics students and we are asked to prove that using the following 6 shapes, The number of ways to build a 2 by N tiling ...
0
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0
answers
35
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Is this generating function correct?
I haven't learned how to calculate generating functions, but I know it has to do with reciprocal functions being equal to regular polynomials, which reminds me of geometric series. I decided to make a ...
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1
vote
1
answer
93
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How can you check if a series$ ∑_1^{\infty} x^{2}e^{−nx}$ is convergent when $ x \in [0,+∞)$? [closed]
How can you check if a series $$∑_1^{\infty} x^2e^{−nx}$$ is convergent when $x$ belongs to $[0,∞)$ step by step? Can D'Alembert's theorem be used? Could someone write it step by step?
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Find the power-series representation for this function.
Here is a practice exam question we've been given (no answer provided; been scratching my head over it for a while):
Here's my work, and answer:
Could anyone verify if this is correct? And then, if ...
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votes
2
answers
76
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2 cars and the fly problem - calculate the total distance covered by the fly
2 cars approach each other, with 20 km between them. The speed of each car is 10 kmph. At 20 km apart from each, a fly starts traveling from one car towards another at 15 kmph. Once it reaches the ...
-4
votes
1
answer
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Finding minimum of geometric sequence
For the given natural number $0<n\in\mathbb{N}$, define the function as follows:
$$f\left(r\right)=2\cdot\sum_{i=0}^{n+2}r^{i-n}+1$$
I want to find the minimum for that function in the domain $r\in\...
0
votes
0
answers
45
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Geometric series summation with power which is also geometric.
I was just pondering how will this series converge. Assuming $n \ge 0$, find
$$
f(n)=n^{1/2} + n^{1/4} + n^{1/8} + n^{1/16} +\text{...}+ 1
$$
We stop the summation when we hit $1$ i.e. $\lfloor{n^{1/...
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2
votes
1
answer
54
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How to prove the following Dirichlet-series/geometric-series idenity, step by step process?
$$\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}$$
What is ...
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votes
1
answer
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Need help understand some simple algebra
I was looking at some text that involves the following equality, where $z \in \mathbb{C}$
$(1+|z|^2+|z|^4+\dots +|z|^{2n-2})=\frac{|z|^{2n}-1}{|z|^2-1}$
I assumed the text used the formula for ...
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1
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Estimation of a series: why/where is my reasoning wrong'
I have to give an estimation of the value of this series
$$\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}
$$
in powers of z to find a series that is valid for an ...
1
vote
1
answer
71
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Is it ok for 'r' to be negative in geometric series?
$6+x+y+z+96 ......$ is a geometric series.
Here we need to find the value of $x$.
Before doing that we need to find the value of $r$.
Here, $a=6$, $ar^4=96$
Now, $r^4=16$, then $r=\pm 2$.
But, a group ...
2
votes
1
answer
49
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Geometric series from -i to i
Is it possible to make 2 geometric series from
$n = -i, -(i-1), -(i-2),...,(i-1),i$
For a finite series I know that a geometric series looks like that,
for $x^n$
$ \sum_0^i = \frac{1-x^{i+1}}{1-x}$
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votes
2
answers
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limit of a geometric-type series
I came across this series while studying probability theory.
Let $$T(x)=\sum_{n=1}^\infty x^n \left(\frac{1}{n}-\frac{1}{n+1}\right),$$
where $T(x)$ converges for some range of $x$.
My question is how ...
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1
answer
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Simplifying a double sum into a single geometric series
I was looking at how to derive the autocorrelation function of a stationary AR(1) process and ran into the following equality in one derivation I found:
$$
\sum_{i=0}^{\infty}\sum_{j=\tau}^{\infty}\...
0
votes
1
answer
65
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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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votes
0
answers
104
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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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vote
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 ...
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3
votes
4
answers
104
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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
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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4
answers
229
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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 ...
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1
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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:
$$\frac {e^{-ajw}...
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2
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Show that $\frac{1-x^n}{1-x}=0\,$ has a real solution at $x$ equals $-1$
I'm trying to solve this question:
$$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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0
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43
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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
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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
14
views
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:
...
0
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1
answer
80
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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
79
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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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0
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35
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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
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29
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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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0
answers
51
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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 $r$ ...
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2
answers
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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 $\alpha \neq \beta$. ...
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1
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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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