Questions tagged [geometric-series]
For questions about or involving geometric series, a series where successive terms have a common ratio.
767
questions
0
votes
2
answers
38
views
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 ...
- 45
2
votes
1
answer
43
views
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 ...
- 123
0
votes
1
answer
39
views
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 \...
0
votes
1
answer
15
views
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 ...
- 149
-1
votes
1
answer
30
views
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:
$a_n = 3*0.5^{n-1}$
$b_n = \frac{3}{2^{n-1}}$
- 23
0
votes
1
answer
31
views
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 ...
- 123
0
votes
1
answer
20
views
"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 ...
- 55
0
votes
0
answers
28
views
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):
$$
P(\lambda) = \...
- 710
0
votes
1
answer
98
views
How radius of convergence of$~{3\over\sqrt{1-9x^2}}~$can be determined as$~{1\over 3}~$?
I want to prove the radius of convergence for the following is$~{1\over 3}~$
$${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}}=\...
- 1,181
2
votes
2
answers
61
views
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 ...
- 123
0
votes
2
answers
58
views
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 ...
2
votes
1
answer
64
views
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 ...
- 1,309
0
votes
0
answers
29
views
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. $...
- 1,507
2
votes
0
answers
55
views
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
...
- 21
0
votes
1
answer
77
views
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 ...
0
votes
0
answers
49
views
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.
[...
- 55
0
votes
0
answers
41
views
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 ...
- 179
-1
votes
1
answer
53
views
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!
...
- 89
4
votes
1
answer
98
views
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 $...
- 525
1
vote
0
answers
59
views
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 ...
- 525
1
vote
0
answers
38
views
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 ...
- 81
17
votes
2
answers
423
views
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:
${\displaystyle {\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots =\sum _{n=1}^{\infty }\left({\frac {1}{2}}\right)^{n}=1.}$...
- 341
0
votes
2
answers
20
views
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
$$\begin{align}
E(X^2)&=\sum_{k=1}^\infty k^2 ...
- 17
-1
votes
1
answer
32
views
Steps to Summation closed form [duplicate]
I have $$\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$$
- 33
5
votes
2
answers
131
views
Confusion regarding infinite series involving complex number
I've been asked to find $\Omega$, such that :
$$\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^...
0
votes
1
answer
26
views
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 ...
- 101
0
votes
2
answers
39
views
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 ...
- 1
3
votes
2
answers
56
views
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
$$
\sum_{n=1}^{N} \mathrm{ln}[A - (n-1)\epsilon ] = N \mathrm{ln}[A - (N-1)\epsilon/2] + \...
- 117
1
vote
1
answer
86
views
Indefinite integral with geometric series
I was trying to calculate the following integral
$$\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} = \...
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 ...
-1
votes
1
answer
31
views
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 -
...
2
votes
2
answers
54
views
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&...
- 105
0
votes
1
answer
31
views
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 $...
- 243
-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.
- 13
1
vote
0
answers
41
views
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
...
- 11
0
votes
0
answers
11
views
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 ...
- 21
0
votes
1
answer
60
views
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 ...
- 19
3
votes
2
answers
59
views
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 $n=x+2$:
$$0.004\sum_{...
- 35
4
votes
1
answer
81
views
Summation problem.
Problem: solve for $x$, such that:$$\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$$
Then solve ...
- 383
2
votes
3
answers
109
views
Polynomial inequality: show $p(x) = \sum_{n=0}^{2k} x^n - \frac{1}{2k+1} \sum_{n=0}^{2k} (-x)^n\ge 0$
Consider the following polynomial in $x$:
$$p(x) = \sum_{n=0}^{2k} x^n - \frac{1}{2k+1} \sum_{n=0}^{2k} (-x)^n.$$
I want to show that $p(x)\geq 0$ for real $x$. It is trivial to show that $p(-1) = 0$, ...
- 23
0
votes
1
answer
93
views
Taylor series for $\frac{1}{(1-x)^{n}}$
So I am trying to find the dimension of the subspace of homogeneous polynomials of degree m $P_m$ of $n$ variables. For $\alpha = (\alpha_1, \dots, \alpha_n$), with $\alpha_1 +\cdots + \alpha_n = m.$
...
- 109
0
votes
2
answers
34
views
sequences and series problem - returns on investment
I am having a problem with these types of problem.
Kenny is offered 2 investment plans , each requiring an initial investment of £10,000.
Plan A offers a fixed return of £800 per year. - arithmetic ...
- 3
0
votes
1
answer
19
views
Manipulation of inferior and superior limits of a summation
Can you tell me if it's ok to manipulate the limit of a summation in this way please?
$$ \sum \limits _{l = k_o}^{+ \infty} q^{l - k_0} = \sum \limits _{l - k_0 = 0}^{+ \infty} q^{l - k_0} $$
Are ...
8
votes
0
answers
195
views
Minimizing a generalized partial sum of a geometric progression
Let $q$ be a rational number satisfying $1/2<q<1,$ so that $\left\{q^n\right\}_{n=0}^\infty$ is a decreasing infinite geometric progression with rational terms. Consider absolute values of ...
- 45.3k
1
vote
1
answer
111
views
How do I separate imaginary and real terms of Cartesian Coordinates of finite complex geometric sum?
Recently, I was looking at the generalized solution for a sum of the geometric series and/or sequence where $r = p+qi$ where $p$ and $q$ are real numbers. I started with the binomial theorem which ...
- 121
0
votes
0
answers
38
views
Find smooth version of $g(x) = \sum_{i=0}^x a^x$
I want to find a smooth function $f$ such that for all positive integers $x$, $f(x)=g(x)$ where $g(x)$ is given below. In less mathy terms I want a smooth version of $g(x)$ constrained by its integer ...
2
votes
1
answer
59
views
Geometric sum to calculate matrix inverses
This is the geometric series sum we all know and love:
$$
\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}
$$
This works ...
- 216
0
votes
0
answers
33
views
Geometric sum of transpose matrices
Is there any mapping between the geometric sum of $A$ and the geometric sum of $A^T$?
To be precise are there conditions under which $(I-A)^{-1}$ is approximately close to $(I-A^T)^{-1}$, other than ...
-2
votes
3
answers
120
views
Calculate $\displaystyle\sum_{n=0}^\infty n \Big( \frac{4}{5}\Big)^{n+1}$ [duplicate]
Exercise Calculate $\displaystyle\sum_{n=0}^\infty n \Big( \frac{4}{5}\Big)^{n+1}$
This is what I have so far
$$\displaystyle\sum_{n=0}^\infty n \Big( \frac{4}{5} \Big)^{n+1} = \displaystyle\sum_{n=0}^...
- 1,117
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 ...
- 11