# Questions tagged [geometric-series]

For questions about or involving geometric series, a series where successive terms have a common ratio.

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### A conceptual misunderstanding in limits and geometric series

So, it starts off like this: My friends told me that the floor function of 0.9 repeating (a.k.a. 0.99999.... ) is 0, which is factually untrue since 0.9 repeating is known and proven to be exactly ...
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### What is the correct notation for geometric sequence?

I'm trying to write a formula that generalizes the following set of equations: $y_1=\sqrt{50-1}$ $y_2=\sqrt{50-2}+y_1$ $y_3=\sqrt{50-3}+y_2$ $y_4=\sqrt{50-4}+y_3$ ... I'm not very familiar ...
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### Find limit of Decrementing Recursive Series

I want to find a formula to find the lower limit part of this recursive or geometric series $$x_{n} = \left( x_{n-1} + p \right) \times \left( 1 - \frac{t}{100} \right)$$ I was just wondering what ...
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### Relation between two series with equal sum of series

In the series S = $\frac{x}{1-x^2}$ + $\frac{x^2}{1-x^4}$ + $\frac{x^4}{1-x^8}$+ ....$\infty$ by solving it by method of difference($V_n - V_{n-1}$) we get S = $\frac{x}{1-x}$. Also, we know that Sum ...
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1 vote
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### Why this does not add up to 0?

i was wondering, why the following sum does not add up to $0$. Consider the following sum of $S_n$ : $$\sum_{n=0}^\infty S_n = S_1 + S_2 + S_3 + ... = \epsilon$$ And the specific elements looks like ...
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### Mean of geometric distribution

I'm trying to prove that $$\sum_{x = 1}^{\infty}x(1 - \pi)^{x - 1}\pi = \frac{1}{\pi}.$$ I've seen elsewhere derivations that involve taking the derivative of a slightly modified version of the above ...
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### Can $1, -1, 1, -1, 1, -1, 1, -1, \dots$ be called a geometric sequence?

The sequence $1, -1, 1, -1, 1, -1, 1, -1, \dots$ seems to satisfy a geometric sequence. But it is an oscillating function. I thought a geometric function should be monotonically increasing or ...
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### Geometric Sequences $A$ and $B$ with Common Ratio; Finding Missing Term in Sequence $C$

$A$ and $B$ are both geometric sequences, and the common ratio of $B$ is $1/3$. $C$ is a sequence created by adding corresponding elements if $A$ and $B$. If $C=\{ 97,51,45,m,... \}$, find the value ...
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1 vote
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### Is there a function $f$ for which the following is not true?

I was working with the geometric series $$1+x+x^2+x^3+\dots=\sum_{n=0}^{\infty}x^n \qquad |x| < 1$$ for which the sum is known to be $$\sum_{n=0}^{\infty}x^n = \frac{1}{1-x} \qquad |x| < 1$$ ...
1 vote
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### Solution or upper bound for "geometric-type" series $\sum_{k=1}^\infty \left(2^k a \,C^{2^k a/2}\right)^{-1}$

How can I find a closed form solution for $\sum_{k=1}^\infty \left(2^k a \,C^{2^k a/2}\right)^{-1}$, for fixed $C>1$ and $a>1$? Clearly the sum is finite. However, I need to evaluate this sum so ...
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1 vote
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### Is sequence $\sum_{k=1}^{n}\sin\left( k+\frac{1}{k}\right)$ bounded? If so, does $\sum_{k=1}^{\infty}\sin k-\sin\left(k+\frac{1}{k}\right)$ converge?

It is well-known that the sequence $a_n:=\displaystyle\sum_{k=1}^{n} \sin k$ is bounded. I want to see if $\displaystyle\sum_{k=1}^{\infty} \sin k - \sin\left( k+\frac{1}{k} \right)\$ converges. The ...
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### Geometric series proof

I was supposed to prove: Theorem: Let $a \neq 0$, the geometric series $\sum_{n=1}^{\infty}(aq^{n-1})$ i) if $|q|< 1$, converges and has the result $S=\frac{a}{1-q}$. ii) if $|q| \ge 1$, diverges. ...
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### Geometric series with indexed inequality

I wanted to complete the following sum: $$\sum_{0\leq i <j<k} a_ib_jc_k$$ Where $a_ib_jc_k$ are all different geometric sequences with $|r|<1$. My attempt was to break up the sum into what ...
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### Prove solution formula of the Water Bottles problem

Problem There are n water bottles that are initially full of water. You can exchange m empty water bottles for one full water bottle. The operation of drinking a full water bottle turns it into an ...
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1 vote
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### Upper bounds on the greatest common divisor of sums of geometric series

Let $S_1=\sum_{i=0}^{n} p^i = \frac{p^{n+1}-1}{p-1}$ and $S_2=\sum_{i=0}^{m} q^i = \frac{q^{n+1}-1}{q-1}$ be two sums of geometric series, and $\gcd\left(S_1,S_2\right)$ its greatest common divisor. ...
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### Variants of geometric sum formula

I know there are closed formulas for sums of the form $\sum_{k=0}^n k^sr^k$ and $\sum_{k=0}^n r^{2n}$ or $\sum_{k=0}^n r^{2n+1}$. (See https://en.wikipedia.org/wiki/Geometric_series#Sum) From Sum of ...
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### Prove minimum of $\sum a_n^2/a_{n+1}$ without using Cauchy Schwarz inequality

If $a_n$ is a decreasing sequence of real numbers and $a_0=1$. How to prove the minimum value of $\sum a_n^2/a_{n+1}$ is 4 without using Cauchy Schwarz inequality? Here's what I got: If $a_n=1/2^n$, ...
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### How would I go about finding the sum of a non-infinite summation?

I am given: $\sum_{k=1}^n 2^{n(1+k)}$ and I am honestly at a loss on how to proceed. I'm thinking to use a geometric series formula, but the index starts at k=1, and there is a "n" in my ...
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### Compute this numerical series

i would like to find a formula that is true for $n \geq 1$ for that : $\displaystyle \sum _{k=1}^n \frac{n^k}{k! \times k}$ I already tried many things like telescoping the sum or symetrising it but i ...
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1 vote
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### Confidence interval for mean based on a single trajectory of a first-order autoregressive process

I am currently studying Statistics for Spatial Data, revised edition, by Cressie. Chapter 1.3 STATISTICS FOR SPATIAL DATA: WHY? says the following: 1.3 STATISTICS FOR SPATIAL DATA: WHY? Some simple ...
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### Solving a functional equation $\alpha f(x + y) = f(x)f(y)$

Suppose a differentiable $f:\mathbb{R}\to(0,∞)$ satisfying $$\alpha f(x+y)=f(x)f(y)\ (\alpha >f(1)).$$ Express the following sum in terms of $\alpha$ and $f(1)$: $$S=\sum_{i=1}^∞ f(i).$$ Method 1:...
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### Number of spins until a bicycle stops

After taking off the feet from the pedal, the front wheel of a bicycle spins 500 times during the first minute. In the next minute, it spins $\dfrac{3}{5}$ of the number it did in the last minute and ...
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### Converting to a familiar form

$(-1)^n × 2^{1/n}$ Is it possible to convert this into the form $ar^{n-1}$? I am not so sure on how to convert this. Can someone give me hints or someone guide me in solving this problem. Additionally,...
1 vote
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### How to justify this estimation?

I was looking (again) at the series $$\sum_{n = 0}^{+\infty} \frac{3^n}{n!}$$ to give an estimation of its sum withouth the knowledge of the exponential series, and following the steps of a ...
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### The sum of the first and second term in a gp is 108 and the sum of the third and fourth term is 12. Find the 2 possible values for the 1st 2 terms

This is a sum from the chapter review section of my book and the answer is 81, 27 or 162, -54. But I didn't understand the process.
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### N-digit geometric numbers which relate to arithmetic progression

Call a $3$-digit number geometric if it has $3$ distinct digits which, when read from left to right, form a geometric sequence. So, consider the number $931$. Let us note $931-792=139$ which means ...
57 views

### Is my reasonment correct? (infinite piecewise series)

Calculate $$\sum_{n = 0}^{+\infty} a_n,$$ where $$a_n = \begin{cases} \frac{1}{2^n} & n\ \text{even} \\\frac{1}{3^n} & n\ \text{odd} \end{cases}$$ What I thought: I split the series into even ...
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### To find the sum of the series

It is a part of log function. please help me to find the sum of this infinite series $$1+e^{2x}+e^{6x}+e^{12x}+e^{20x}+\ldots$$ or $$\sum_{n=1}^{\infty} e^{n(n-1) x}$$ I got this problem while ...
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### Show decimal expansions

I can't wrap my head around this exercise: Show that the rational number $\frac 94$ has two different decimal expansions, namely $2.2500000\dots$ and $2.2499999\dots$ by writing these decimal ...
1 vote
56 views

### Operational Calculus: $\left ( 1-D \right )^{-1} x^3$ expanded as the geometric series $\left ( 1+D+D^2+D^3+\cdot \cdot \cdot \right )x^3$

While watching a YouTube video by Supware, titled "The Abstract World of Operational Calculus", the speaker is working with the derivative operator $D$. He says he will "formally" ...
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### Series where the sum is 0 [closed]

I'm not a mathematician and don't know a lot about series but I'm trying to figure out if there is an infinite geometric series that starts at 1 (or any other positive number), has fractions always ...
1 vote
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1 vote
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### Can the sums of the first few terms of two geometric progressions with different prime common ratios (start from $1$) be equal?

Consider two geometric progressions with different prime ratios, which both start from $1$. The question is, can the sum of the first $m$ terms of one progression equal to the sum of the first $n$ ...
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### given that the sum of the first n terms of a geometric sequence is $4-(2^{(n+2)})/3^{(2n)}$. find the second term and its common ratio. [duplicate]

I tried to find the first term by substituting 1 into n, and I got $28/9$. After that, I subtitute $2$ into $n$ for $S_n$ and dis the $Term_2 = S_2 -S_1$. I wasn't sure whether I am correct or not. ...
64 views

### Squares and geometric series [duplicate]

Recently I've interested in the question: What is the relation between squares and sum of geometric series? In particular, I'm studying the following diophantine equation: $a^n -1 = (a - 1) m^2$ I'm ...
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### For which $x,n$ is the finite geometric series a perfect power?

Let $x,n,y,q$ be integers greater than one with $n>2$ as well (this is to avoid trivial solutions). The closed form for the geometric series $S(x,n)=\sum_{k=0}^{n-1} x^k$ is $\frac{x^{n}-1}{x-1}$. ...
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### Geometric series with binomial coefficients

Please sum this sum , $\sum_{k=0}^{n} \binom{r+a+bk}{a+bk} x^{bk}$. Where r ,a and b are fixed integers
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