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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Quick formula for $f(n) = \sum_{i=0}^{i=n}{k^{ji}}$ where $k \in \boldsymbol{\mathbb{R}}$ & $j \in \boldsymbol{\mathbb{N}}$

I am trying to figure out a function $f(n)$ which takes the input $n$, where $n \in \boldsymbol{\mathbb{N}}$, and outputs the sum $\sum_{i=0}^{i=n}{k^{ji}}$ till the $n^{th}$ value where $k \in \...
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Prove $|\sum_{j=0}^n \frac{5^j}{c^j}-\frac{c}{c-5}|\leq\frac{(5^{n+1})}{|c|^n(|c|-5)}$

Determine whether the statement is true or false. Let $c\in C,$ and $n\in \Bbb N$ \ {$0$}. Suppose $|c|\gt5$. Then $|\sum_{j=0}^n \frac{5^j}{c^j}-\frac{c}{c-5}|\leq\frac{(5^{n+1})}{|c|^n(|c|-5)}$. As ...
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Are two geometric sequences with the first $a_1$ and the same common ratio identical?

If I have two geometric sequences, and they both have the same first term and the same common ratio, can I say they are the same? For example: $a_n = 3*0.5^{n-1}$ $b_n = \frac{3}{2^{n-1}}$
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Find the values $p$ and $q$ for when the geometric series converges [closed]

The numbers $p, 10, q$ are the consecutive terms of an arithmetic series. The numbers $p, 6, q$ are from a geometric series. Show that $p^2-20p+36=0$ and hence find the values of $p$ and $q$ for which ...
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"modified" geometric series

Good evening, Let $n, b \in \mathbb{N}$, $n,b \geq 2$ and $b \in \mathbb{N}$, then we know that $\sum_{k=0}^n b^k = \frac{b^{n+1}-1}{b-1}$. In other words we have $\sum_{k=0}^n b^k = c_{1,b} \cdot b^n ...
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Do Polylogarithms Always Converge

I have been reading the Wikipedia page dedicated to polylogarithms to understand the following paper. I am trying to understand the first term in equation (6) (reproduced below): $$ P(\lambda) = \...
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How radius of convergence of$~{3\over\sqrt{1-9x^2}}~$can be determined as$~{1\over 3}~$?

I want to prove the radius of convergence for the following is$~{1\over 3}~$ $${3\over\sqrt{1-9x^2}}\tag{1}$$ By the advice from@Kavi Rama Murthy, I thought the following. $$f(y):={1\over\sqrt{1+y}}=\...
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2 answers
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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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1 answer
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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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In real numbers, does the series converging to them have a generating sequence?

From a sequence: $a,b,c,d,e,f....$ we get another sequence which we call series with successive addition i.e. $a, a + b, a + b + c, a + b + c + d....$ Now if we have a periodic decimal fraction e.g. $...
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Condition for two quadratic equation to have one common root (Simplification)

If a,b,c are in Geometric Progression, then the equations $ax^2+2bx+c=0$ and $dx2+2ex+f=0$ have a common root if $\frac da, \frac eb, \frac fc$ are in: Arithmetic Progression Geometric Progression ...
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Finding the general formula for the sum of the first n terms.

So I'm trying to solve this practice problem I found on the internet. PracticeProblemLink $$\displaystyle{ \sum_{n=0}^{\infty} {2e^{-n}} }$$ The part I'm stuck on is this: The general formula for the ...
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Taylor Series of complex numbers

I think that I would use the Taylor series but I'm not sure where to start. I know that that Taylor series is equal to $$ \frac{f^{(n)}(p)}{n!} \cdot (z-p)^n $$ but I'm not sure how to use this fact. [...
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The geometric distribution - How many trials occur before we obtain a success doubt

I have doubts on the same question First, What is k here? Second, I know E[X]= E[I{A}], I define indicator random variable. We ...
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Is there a geometric sequence that contains each of the numbers 1, 2 and 3? [closed]

I recently got the question: Is there a geometric sequence that contains each of the numbers 1, 2 and 3? I have tried my best to make a start with the $x_n=ar^{(n-1)}$ theorem but I'm still puzzled! ...
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4 votes
1 answer
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Are Cantor-like sets disjoint for $\xi,\eta$ with no common power?

Let $\xi,\eta \in (0,\frac{1}{2})$. Let $C_\xi$ (and analogously for $C_\eta$) be the perfect symmetric set built by iterating the transformation $$[0,1] \to [0,\xi]\cup [1-\xi, 1].$$ Will the sets $...
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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 ...
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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: ${\displaystyle {\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots =\sum _{n=1}^{\infty }\left({\frac {1}{2}}\right)^{n}=1.}$...
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Problem understanding a step in the calculation of the variance of the geometric random variable

I am following the following book to study statistics. I have the demostration of the calculation of the variance of the geometric random variable as $$\begin{align} E(X^2)&=\sum_{k=1}^\infty k^2 ...
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1 answer
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Steps to Summation closed form [duplicate]

I have $$\sum_{n=1}^k 2^n$$ I got this result from trial and error (validated online), but I want to understand the steps to get there. $$= 2^{k+1}-2$$
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5 votes
2 answers
131 views

Confusion regarding infinite series involving complex number

I've been asked to find $\Omega$, such that : $$\Omega=\frac{\Omega_1}{\Omega_2}=\frac{1+e^{\frac{2i\pi}{3}}+e^{\frac{4i\pi}{3}}+e^{\frac{6i\pi}{3}}+e^{\frac{8i\pi}{3}}+.....}{i+\frac{i^2}{2}+\frac{i^...
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1 answer
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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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3 votes
2 answers
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Approximating the summation of the logarithm of a function

I am working through a textbook for self-study and came across an approximation that I'm unsure how to solve $$ \sum_{n=1}^{N} \mathrm{ln}[A - (n-1)\epsilon ] = N \mathrm{ln}[A - (N-1)\epsilon/2] + \...
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1 vote
1 answer
86 views

Indefinite integral with geometric series

I was trying to calculate the following integral $$\int \frac{\text{d}x}{(1 + x^n)^n}$$ for $n > 0$. I tried this road: $$\int \frac{\text{d}x}{\left(x^n\left(1 + \frac{1}{x^n}\right)\right)^n} = \...
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1 vote
3 answers
65 views

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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-1 votes
1 answer
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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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2 votes
2 answers
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What is the supremum of $\frac{\sum_{i=1}^\infty |x_i|2^{-i}}{\|x\|_p}$?

I have been trying to find the upper limit of this of the of elements $$\frac{\sum_{i=1}^\infty |x_i|2^{-i}}{\|x\|_p}$$ such that $x \in l^p$, i.e $\sum_{i=1}^\infty |x_i|^p < \infty$ and $ 1<p&...
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Is the system is a geometric progression?

I am trying to show that the following system is a geometric progression i.e. $a_2=a_1^2$, $a_3=a_1^3$ etc. $a_{k}^{2}=a_{k-1}a_{k+1}$ $a_{k-1}^{2}=a_{k-2}a_{k}$ $\cdots$ $a_{2}^{2}=a_{1}a_{3}$ here $...
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-1 votes
1 answer
87 views

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.
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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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0 answers
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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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1 answer
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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 ...
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3 votes
2 answers
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Finding the sum of geometric progression

Evaluate the sum: $$\sum_{x=0}^\infty x(x-1) {2+x \choose x}(0.008)(0.8)^x $$ I was able to make this into: $$0.004\sum_{x=0}^\infty x(x-1) (x+1)(x+2)(0.8)^x $$ Let $x=n-2$ then $n=x+2$: $$0.004\sum_{...
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4 votes
1 answer
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Summation problem.

Problem: solve for $x$, such that:$$\sum_{n=0}^{\infty } x^n=\dfrac{4}{11}$$ My first idea was to evaluate the series: $$\sum_{n=0}^{\infty } x^n=\dfrac{1}{1-x}, \text{ for }|x|<1$$ Then solve ...
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2 votes
3 answers
109 views

Polynomial inequality: show $p(x) = \sum_{n=0}^{2k} x^n - \frac{1}{2k+1} \sum_{n=0}^{2k} (-x)^n\ge 0$

Consider the following polynomial in $x$: $$p(x) = \sum_{n=0}^{2k} x^n - \frac{1}{2k+1} \sum_{n=0}^{2k} (-x)^n.$$ I want to show that $p(x)\geq 0$ for real $x$. It is trivial to show that $p(-1) = 0$, ...
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0 votes
1 answer
93 views

Taylor series for $\frac{1}{(1-x)^{n}}$

So I am trying to find the dimension of the subspace of homogeneous polynomials of degree m $P_m$ of $n$ variables. For $\alpha = (\alpha_1, \dots, \alpha_n$), with $\alpha_1 +\cdots + \alpha_n = m.$ ...
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0 votes
2 answers
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sequences and series problem - returns on investment

I am having a problem with these types of problem. Kenny is offered 2 investment plans , each requiring an initial investment of £10,000. Plan A offers a fixed return of £800 per year. - arithmetic ...
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0 votes
1 answer
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Manipulation of inferior and superior limits of a summation

Can you tell me if it's ok to manipulate the limit of a summation in this way please? $$ \sum \limits _{l = k_o}^{+ \infty} q^{l - k_0} = \sum \limits _{l - k_0 = 0}^{+ \infty} q^{l - k_0} $$ Are ...
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8 votes
0 answers
195 views

Minimizing a generalized partial sum of a geometric progression

Let $q$ be a rational number satisfying $1/2<q<1,$ so that $\left\{q^n\right\}_{n=0}^\infty$ is a decreasing infinite geometric progression with rational terms. Consider absolute values of ...
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1 vote
1 answer
111 views

How do I separate imaginary and real terms of Cartesian Coordinates of finite complex geometric sum?

Recently, I was looking at the generalized solution for a sum of the geometric series and/or sequence where $r = p+qi$ where $p$ and $q$ are real numbers. I started with the binomial theorem which ...
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0 answers
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Find smooth version of $g(x) = \sum_{i=0}^x a^x$

I want to find a smooth function $f$ such that for all positive integers $x$, $f(x)=g(x)$ where $g(x)$ is given below. In less mathy terms I want a smooth version of $g(x)$ constrained by its integer ...
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2 votes
1 answer
59 views

Geometric sum to calculate matrix inverses

This is the geometric series sum we all know and love: $$ \sum^\infty_{n=0} a^n=\frac{1}{1-a} $$ By moving things around a little bit we get: $$ \sum^\infty_{n=0} (1-a)^n=\frac{1}{a} $$ This works ...
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0 answers
33 views

Geometric sum of transpose matrices

Is there any mapping between the geometric sum of $A$ and the geometric sum of $A^T$? To be precise are there conditions under which $(I-A)^{-1}$ is approximately close to $(I-A^T)^{-1}$, other than ...
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-2 votes
3 answers
120 views

Calculate $\displaystyle\sum_{n=0}^\infty n \Big( \frac{4}{5}\Big)^{n+1}$ [duplicate]

Exercise Calculate $\displaystyle\sum_{n=0}^\infty n \Big( \frac{4}{5}\Big)^{n+1}$ This is what I have so far $$\displaystyle\sum_{n=0}^\infty n \Big( \frac{4}{5} \Big)^{n+1} = \displaystyle\sum_{n=0}^...
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1 vote
1 answer
261 views

Sum of series (bouncing ball)

When dropped, a ball takes 1 second to hit the ground. It then takes 90% of this time to rebound to its new height, and this continues until the ball comes to rest. a) Show that the total time of ...
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