Questions tagged [geometric-series]
For questions about or involving geometric series, a series where successive terms have a common ratio.
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1answer
55 views
How to solve this sum $\sum_{n=0}^{\infty} e^{-0.5n(n+1)}$ [on hold]
For my homework I need to solve the sum $$\sum_{n=0}^{\infty} e^{-0.5n(n+1)}$$
I tried to rewrite it as a geometric series because it kind of looks like one but to no success.
1
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0answers
46 views
Converting an Arithmetic Series to Sigma Notation
I've been struggling with the following problem for quite a while now, and have been unable to identify a pattern;
You have a geometric series $Y$ for which we have the following rule:
$$Y_{t+1} ...
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0answers
13 views
Expression for the sum of another bounded series with arithmetic and geometric components
The expression I have is the following:
$S = \sum_{i=1}^T (1-q)^{i-1}i$
My expression after I play around with it is:
$S= \frac{1-(1-q)^TT}{q}+\frac{1-q}{q} \left(\frac{1-(1-q)^{T-1}}{q} \right) $...
0
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2answers
24 views
Relation between two series
Consider the two series ,
A=Σ(2ⁿ/n!) from 1 to ∞.
and,
B=Σ(4ⁿ/n!) from 1 to ∞.
What is the relationship between them?( If any)
I think the exponential series might come in handy but the numerator ...
1
vote
1answer
37 views
sums of series involving complex numbers
I dont know how to use latex any good rescources to quickly learn from scratch, hence why I have photographed the question.
The question is as follows
question
question final part of working
The ...
0
votes
1answer
42 views
Simplify a finite series [closed]
can someone help me simplify the following formula?
$$y = \pi(x-a)-\frac{(\pi(x-a))^3}{3!}+\frac{(\pi(x-a))^5}{5!}-\frac{(\pi(x-a))^7}{7!}+\frac{(\pi(x-a))^9}{9!}-\frac{(\pi(x-a))^{11}}{11!}+\frac{(\...
2
votes
0answers
45 views
What does it mean to say it “converges geometrically”
I've been reading up on some algorithms and I've come across the statement "it converges geometrically" but what does this mean? How does it differ from saying "it converges"? Is there some "rate" ...
0
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1answer
33 views
Can a geometric sequence go on forever?
I have a geometric sequence whose n-th term is $11^n$:
$1,11,121,1331,...$
I want to know if the pattern continues forever. I understand that if I sum these numbers up, my common ratio is 11 which ...
0
votes
1answer
25 views
Explanation for visual representation of geometric series
I found these on the Art of Problem Solving website here.
I just don't see how the first image would give you $\frac{1}{3}+\frac{1}{3^2}+...$ and I'm unable to see how the second one is divided into ...
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0answers
14 views
Absolute convergence of series variant of the geometric series
So I want to prove whether the following series converges absolutely or not:
$$\frac{1}{2}\sum_{n=0}^\infty (n^2+3n+2)q^n$$
where $ q \in \mathbb{C}, \mid q\mid<1.$
My attempt was:
$$\frac{1}{2}...
0
votes
1answer
41 views
Cauchy product and geometric series [duplicate]
I was given this series:
Let $q \in \mathbb{C}, \mid q\mid <1. $
$$\frac{1}{2}\sum_{n=0}^{\infty} (n^2 +3n +2)q^n $$
Now I have to show that
$$\frac{1}{(1-q)^3}=\frac{1}{2}\sum_{n=0}^{\infty} (...
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votes
1answer
12 views
comparing area of a square with area of rectangle
If we have a geometric sequence we have $a_1, a_1*r, a_2*r$
So $b=a_1*r$ and $c=a_2*r$
The square has area $b^2$ which is $(a_1*r)^2$ and the rectangle has area $ac = a_1*a_2*r$
I don't know ...
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4answers
38 views
Sum of “intertwine” of Arithmetic & Geometric Series (to infinity)
I am just wondering is there any method to evaluate the following:
$$
2\cdot (1/2) + 3\cdot (1/4) + 4\cdot (1/8) + 5\cdot (1/16) + \ldots =\sum_{n=2}^\infty \frac{n}{2^{n-1}}
$$
Frankly, I have no ...
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1answer
64 views
Working with the coupon collector problem
Here is the problem I have:
Say that each box of cereal contains a prize. There are $n$ different prizes in all.
What is the probability that more than $t$ boxes are needed to find all $n$ gifts?
...
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2answers
27 views
Calculating finite geometric sum
What is the sum: $\displaystyle\sum_{n=0}^{31} 2^n$?
I know the formula is $S_n = \frac{a_1 (1-r^n)}{1-r}$ , so I have
$$S_n = \frac{1(1-2^{32})}{1-2} = \frac{-4294967295
}{-1} = 4294967295$$
Is ...
2
votes
1answer
31 views
Deriving complicated identity
I'm in need of help deriving this identity (due to Ramanujan):
$$\sum _{n=0}^{\infty } \frac{\left(a^{n+1}-b^{n+1}\right) \left(c^{n+1}-d^{n+1}\right)}{(a-b) (c-d)}T^n=\frac{1-a b c d T^2}{(1-a c T) (...
3
votes
3answers
57 views
need help solving geometric series questions
A geometric sequence has its first term as $10000$ and a fourth term as $−7290$.
If the pattern continues forever, find the sum of the terms in the sequence.
I know that the $n^{th}$ term is found ...
0
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3answers
45 views
Find the sum of golden ratios in geometric series
First: Do i need to determine whether these sums converge or diverge?
My attempt:
$\sum_{n=0}^{\infty} (\frac{1}{\phi})^n$
$r = \frac{1}{\phi}$
sum is then $\frac{1}{1-\frac{1}{\phi}}$
$\sum_{n=...
0
votes
1answer
26 views
find the ratio of amount of rice placed on a chessboard between first and second half of chessboard?
I think the sequence for this is:
1, 2, 4, ...
I'm not sure what it means by first half of chessboard vs next half of chessboard. I'm assuming it is some kind of ratio?
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votes
1answer
29 views
Finding the smallest possible value of the sum in infinite geometric sequence and another geometric sequence question
The sum of an infinite geometric series is a positive number $S$, and the second term in the series is $1$. What is the smallest possible value of $S$?
For this I am thinking the $t_1, 1, t_2,...$
$...
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3answers
44 views
Questions on geometric series
Consider a three-term arithmetic sequence whose terms are $t_1, t_2, t_3$. The related sequence $t_1+8, t_2,t_3,$ is geometric and the sum of its terms is 26. Find the terms in the sequence
For the ...
0
votes
3answers
57 views
Exact calculation of $\sum_{n=0}^\infty \cos(36^\circ)^n$
Right now we are covering geometric series.To find $\cos 36^\circ$ I use the sum and difference formula.
Knowing that $\cos36^{\circ} = \frac{1+\sqrt{5}}{4} = \phi/2$ calculate $\sum_{n=0}^{\infty}\...
0
votes
1answer
23 views
Establishing an inequality between the first term of an infinite geometric series and the infinte sum?
An infinite geometric series has the first term a and sum to infinity b, where b $\neq 0$. Prove that a lies between 0 and 2b.
$
\rightarrow \text{Since the series converges, } r \text{ has to be ...
4
votes
3answers
249 views
Finding an Explicit Formula for a Geometric Series
CONTEXT: Uni question made up by lecturer
If you have to take a $100$mg drug every $8$ hours, and just before you take the drug, $20$% of it remains in your body, how would you write an explicit ...
0
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0answers
39 views
Sum involving powers and a constant term
Let $c$ be a positive real constant. What is the value of the sum $\sum\limits_{k = 1}^{\infty} \dfrac{1}{c + 4^k}$ in terms of $c$. Closed form solutions please
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1answer
35 views
Is there a list of formulas for geometric series?
For example:
$1 + 2 + 3 + ... + n = \frac{n(n+1)}{2}$
$1+x+x^2+...+x^{n-1}=\frac{x^n-1}{x-1}$
I'm sure there are many more but there doesn't seem to be any source that has a list of these formulas.
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3answers
40 views
How do I find the common ratio of a geometric sequence
A geometric sequence has its first term equal to 12 and its fourth term equal to -96.
How do I find the common ratio
And find the sum of the first 14 terms
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1answer
19 views
How can I determine the form of a geometric series in a recurrence relation by using iterative substitution?
This question is based more in computer science however I think a mathematics approach is probably better. When solving a recurrence relation using iterative substitution you generally need to find ...
6
votes
1answer
70 views
Show uniform convergence and pointwise convergence for $\sum_{n=1}^ \infty \frac{z^ {n-1}}{(1-z^n)(1-z^ {n+1})}$
Consider the series:
$$\sum_{n=1}^ \infty \frac{z^ {n-1}}{(1-z^n)(1-z^ {n+1})}$$
show this converges to:
(a) $\frac{1}{(1-z)^2}$ for $|z|<1$
(b) $\frac{1}{z(1-z)^2}$ for $|z|>1$
...
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1answer
43 views
Switching Integral and Sum
I want to proof that I can switch this Sum and Integral
$\sum\limits_{n=1}^\infty\int\limits_{0}^\infty t^{z-1} e^{-nt}dt~~$ for $~ 1 < Re(z) $
to sum it after over n. I tried to use the ...
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vote
2answers
56 views
Summation of ${\sum_{k=1}^n k\binom{n}{k} = n2^{n-1}}$ using equations
I was trying to solve the summation:
$${\sum_{k=1}^n k\binom{n}{k} = n2^{n-1}}$$
I started something like:
$${\sum_{k=0}^n \binom{n}{k} = 2^n}$$
$${\Rightarrow \sum_{k=1}^n \binom{n}{k} = 2^n - 2^...
2
votes
0answers
25 views
Is this a correct Laurent series expansion for the given annulus?
Expand the function
$$f(z) = \frac{1}{(z + 1)(z + 3)}$$
in a Laurent series valid for $1 < |z| < 3$
My attempt:
$$\frac{1}{(z + 1)(z + 3)}=\frac{1}{4}.\frac{1}{1+z}-\frac{1}{4}.\frac{1}{3+z}$$
...
0
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2answers
25 views
finite geometric series ,2 different formulas?
Are both formulas the same?Do they have different use?Is there something else i am missing here? I do not ask for the proof of the formulas.
i see both of them online on various resources, for ...
1
vote
2answers
58 views
Is the sum of the series $\sum 1/3^n$ equal to $1/2$ or $3/2$?
The formula for a geometric series as I know it is $\sum ar^{n-1}$, where $r$ is the common ratio and $a$ is the first number the common ratio is multiplied with.
If we're to conform to that formula,...
0
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1answer
27 views
Laurent Series of rational function in z
Im attempting to solve a problem of defining the types of singularities of a complex function. I found that if the function has a Laurent series expansion with a finite number of terms in the ...
0
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0answers
28 views
When does the geometric limit hold for matrices?
We know that for scalars, $a+ax+ax^2+...\infty=\frac{a}{1-x}$ provided $|x|<1$. Suppose $x$ were a square matrix. What are the necessary conditions for $aI+ax+ax^2+...=a(I-x)^{-1}$ to hold?
My ...
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2answers
71 views
Find $f^{(22)}(0)$ for $f(x)=\frac{x}{x^{3}-x^{2}+2x-2}$
Find $f^{(22)}(0)$ for $f(x)=\frac{x}{x^{3}-x^{2}+2x-2}$
I know that I should use Taylor's theorem and create power series. However I don't have idea how I can find $a_{n}$ such that $f(x)=\sum_{n=1}^...
1
vote
2answers
43 views
Can a sum of a finite number of exponentially growing numbers be calculated as a function of the growth rate and the number of growths?
This is a question based on the exponential growth pattern of a game mechanic in World of Warcraft: The Heart of Azeroth item in the game has a level that can be increased by gathering Azerite, with ...
5
votes
5answers
174 views
Interpreting images representing geometric series
I understand the formula for infinite geometric series as
$$S = \frac{a_{1}}{1-r}$$ if $0<r<1$
However I'm having trouble applying it to these images
It seems to me that in the first image, ...
0
votes
0answers
24 views
Probability of a geometric series greater than 0.1 after T trials
Given a random variable which can take on a discrete set of values x = [-0.1 -0.05, 0, 0.05, 0.1]. These values have a discrete set of probabilities p = [0.05, 0.20, 0.50, 0.20, 0.05].
Define the ...
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votes
1answer
38 views
How to find the closed form of the summation below without changing the lower and upper bound of summation? [closed]
The lower bound of summation is i=0 and the upper bound of summation is log(n) - 1 (log is base 2).
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0answers
14 views
Proving the norm expression using Neumann expansion
I have been trying to solve the problem, but wasn't able to move after applying Neumann expansion. The square matrix F satisfies $||F||<1$.
I am trying to show the following.
$$
||(I-F)^{-1}|| \...
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1answer
38 views
Does the $a$ in the definition of a geometric series ($\sum_{n=1}^{\infty} ar^{n-1}$) have to be a real number? Can it incorperate n?
I've found my textbook resorting to things like the ratio test with, for example, the series $$\sum_{n=0}^{\infty} x^{n}/n!$$
Could that series not be considered a geometric series, where $a = 1/n! $ ...
0
votes
1answer
33 views
Geometric Series to Solve for Year a Resource Will Be Depleted
The original problem is as follows:
A community has 300 million tons of a non-renewable resource. Annual consumption is 25 million tons per year. Consumption is expected to decrease by 10% each ...
0
votes
2answers
20 views
Sum of Special Series involving exponents
What is the sum to $n$ terms of the series $$2(2^0)+ 3(2^1) + 4(2^2) + \cdots$$
My try:- The $n$th term is $$(n+1)(2^{n-1}) = n(2^{n-1}) + 2^{n-1}.$$ So the sum is the summation of these two terms. I ...
0
votes
2answers
71 views
Conditional probability greater than 1. Why?
Let's say I roll a fair die independently many times. Let $X_i$ be the outcome of the $i$th roll. Assume that on the $k$th roll, I get a $1$ or $X_k = 1$ (Edit based on comments: assume this is the ...
0
votes
2answers
46 views
Sum of (Almost) Infinite Geometric Series
I have recently stumbled upon this old problem of proving that
$$
\displaystyle\sum_{k=0}^{\infty} \left\lfloor\dfrac{n+2^k}{2^{k+1}}\right\rfloor=n, \, \forall n\in \mathbb{Z}_+
$$
where $\left\...
0
votes
0answers
44 views
Non constant geometric sequence problem?
A problem on the national maths competition which I can't solve no matter how hard I try. It's high school maths so it shouldn't be that difficult. So here is the question:
How many prime numbers can ...
1
vote
1answer
40 views
Formula for this diminishing function
I'm trying to write a function for a game where points accumulated becomes less effective as they get more.
Input Value: 2000
Interval: 500
For every interval reached, decrease the ...
0
votes
1answer
38 views
Compute the Sum of geometric series
Basically, I just came out of my exam and this was a question that I wasn't sure of how to solve, I'd appreciate it if someone answered this
$$\sum_{k=1}^{99}\frac{1}{k(k+1)}$$
