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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 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.
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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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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) $...
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2answers
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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 ...
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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 ...
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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{(\...
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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" ...
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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 ...
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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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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}...
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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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1answer
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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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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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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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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 ...
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1answer
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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) (...
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3answers
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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 ...
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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=...
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1answer
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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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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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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 ...
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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}\...
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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 ...
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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 ...
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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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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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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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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 ...
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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
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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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2answers
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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^...
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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: $$\frac{1}{(z + 1)(z + 3)}=\frac{1}{4}.\frac{1}{1+z}-\frac{1}{4}.\frac{1}{3+z}$$ ...
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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 ...
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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,...
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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 ...
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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}^...
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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 ...
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5answers
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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, ...
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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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1answer
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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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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! $ ...
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1answer
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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 ...
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2answers
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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 ...
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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 ...
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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\...
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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 ...
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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 ...
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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 $$\sum_{k=1}^{99}\frac{1}{k(k+1)}$$