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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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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 ...
32 Bit's user avatar
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0 answers
23 views

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 ...
multipurpose_surface's user avatar
0 votes
2 answers
21 views

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 ...
MrShoe's user avatar
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1 answer
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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 ...
Chetan's user avatar
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0 answers
99 views

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 ...
Balazs's user avatar
  • 19
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1 answer
58 views

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 ...
Riccardo Iorio's user avatar
-2 votes
2 answers
93 views

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 ...
user67275's user avatar
  • 135
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1 answer
20 views

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 ...
David Ma's user avatar
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1 answer
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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 $$\sum_{n=0}^{\infty}x^n = \frac{1}{1-x} \qquad |x| < 1$$ ...
Sifiso Rimana's user avatar
1 vote
1 answer
40 views

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 ...
Chad Brown's user avatar
1 vote
1 answer
77 views

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 ...
Adam Rubinson's user avatar
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1 answer
44 views

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. ...
Ruan Carlos's user avatar
0 votes
1 answer
36 views

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 ...
beigespectacles's user avatar
3 votes
2 answers
104 views

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 ...
iloveseven's user avatar
1 vote
0 answers
92 views

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. ...
Juan Moreno's user avatar
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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 ...
Irwin Shure's user avatar
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0 answers
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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 $a_n=1/2^n$, ...
HIH's user avatar
  • 419
-1 votes
1 answer
43 views

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 ...
elguero's user avatar
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1 answer
45 views

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 ...
Gss's user avatar
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1 vote
1 answer
114 views

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 ...
The Pointer's user avatar
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2 votes
1 answer
110 views

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:...
Cognoscenti's user avatar
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0 answers
20 views

Computationally evaluating messy symbolic sums involving geometric series

Let $t$ be a positive integer, let $p$ be a prime number, and let $q$ be a real number. I need to evaluate the sum $$ \sum_{\substack{1 \leq c \leq t \\ c \not \equiv 1 \pmod{p}}} q^{-\big((p-2)c + \...
Sebastian Monnet's user avatar
2 votes
1 answer
167 views

Leibniz's 1684 Solution to De Beaune's Problem

In his 1684 text, "A New Method for Finding Minima and Maxima," Leibniz solves an inverse-tangent problem posed by De Beaune that basically asks what kind of curve will have a constant ...
MrMagoo's user avatar
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3 votes
1 answer
72 views

Solving the recurrence equation : $T(n) = 2T(n/2) + n^2$

Where $T(1)=1$ and assuming that $n=2^k$ and that $k\ge 0$ and that the Master Theorm can't be used. What I tried: $T(n) = 2T(n/2)+n^2$ Following backwards substitution to get a pattern: $T(2^k)=2T(2^{...
MM7654DDD's user avatar
0 votes
0 answers
23 views

Find the value of lambda for which the error term is negative

I consider the following error term where $\lambda\in(0,1)$. I want to find the value of $\lambda$ for which it is negative $$ \lambda^{n-1} -2\sum_{k=n}^{\infty}\lambda^k $$ We have that $$ \lambda^{...
G2MWF's user avatar
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0 votes
0 answers
22 views

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 ...
mvfs314's user avatar
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0 votes
0 answers
36 views

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,...
salierii's user avatar
1 vote
3 answers
75 views

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 ...
Heidegger's user avatar
  • 3,363
-1 votes
1 answer
58 views

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.
Sky's user avatar
  • 7
0 votes
0 answers
32 views

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 ...
Mikhail Gaichenkov's user avatar
0 votes
1 answer
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 ...
Heidegger's user avatar
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0 votes
1 answer
110 views

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 ...
Arun Kotagi's user avatar
2 votes
1 answer
86 views

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 ...
Matteo Bernasconi's user avatar
1 vote
0 answers
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" ...
JPPM's user avatar
  • 21
0 votes
1 answer
91 views

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 ...
harris 's user avatar
1 vote
1 answer
30 views

Triple summation geometric

I'm trying to solve the following triple summation, but I'm not sure this is the best way to solve it, here's what I tried: $$\sum_{k=3}^{n} \sum_{i=2}^{k-1}\sum_{j=0}^{i} 3^{i+j+k}$$ $n \in \mathbb{...
William's user avatar
  • 95
0 votes
0 answers
94 views

Definite integral of ln(x) from 1 to a as the limit of a Riemann sum

For $a > 1$ determine the definite integral $$ \int_1^a ln(x) $$ as the limit of a Riemann sum. Hint: Use the partitioning $P_N = (x_0, x_1, …, x_N)$ with: $$ 1 = x_0 < x_1 < … < x_N = a \...
sagan's user avatar
  • 1
2 votes
1 answer
50 views

Infinite sum of products identity

Let $\beta,\omega\in(0,1)$ and $z>0$ then the following seems to be true: $$\sum_{j=0}^{\infty}{\beta^j \frac{\prod_{k=0}^{j}{\frac{1+z(\beta\omega)^{-k}}{1+z(\beta\omega)^{1-k}}}}{1+z(\beta\omega)^...
cfp's user avatar
  • 685
0 votes
1 answer
83 views

Solve a geometrico-harmonic infinite sum

I am trying to calculate the infinite sum of a geometric sequence times a harmonic sequence, so a series formatted in this way: $$ \sum_{n=1}^\infty \frac{1}{n}\cdot q^n $$ I simply know that it is ...
Vito Palmieri's user avatar
0 votes
1 answer
39 views

Solve For Multiplying Factors In Geometric Sequence?

I have a sequence of numbers like this one: $a_n = \{1, 10, 66, 406, 2454, \dots\}$ This sequence is not on OEIS, but I know that this is a geometric sequence of this form: $a_n = b_1 A_1^n + b_2 A_2^...
JavamonkYT's user avatar
1 vote
0 answers
50 views

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 $n$ ...
ZhouYang's user avatar
-1 votes
1 answer
311 views

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. ...
tanya smith's user avatar
-1 votes
1 answer
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 ...
dajfufjrgfkdlw dgdvrbkwkslchrj's user avatar
6 votes
0 answers
148 views

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 $\frac{x^{n}-1}{x-1}$. ...
Integrand's user avatar
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1 vote
0 answers
129 views

Why can we crank up the variable $\lambda$ in perturbation theory?

In the formulation of perturbation theory, we use the power series (more specifically geometric series) for small expansion, say $\lambda$, to expand the exact case, say $H^0$, so $\begin{aligned} &...
Rasmus Andersen's user avatar
0 votes
0 answers
31 views

Count of x in interval [l; r]

I have $l$ and $r$ and I have to find the number of such integers $x$ $(l ≤ x ≤ r)$, that the first digit of integer $x$ equals the last one (in decimal notation). For example, such numbers as $101$, $...
Isam's user avatar
  • 67
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0 answers
58 views

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
Brajesh Kumar Dhakad's user avatar
5 votes
1 answer
103 views

Summation of an Arithmetico-Geometric Progression

$$ \sum_{r=1}^{11} r5^r = \frac{43\times5^{a}+ 5}{b}$$ Find (a+b) This question was asked in an exam. I got the answer 28. However the answer given was 15. Here is my attempt: Let S = $ \sum_{r=1}^{11}...
BlackHood's user avatar
  • 171
3 votes
1 answer
46 views

Geometric distribution over the natural numbers

I’m thinking of the geometric distribution over $\mathbb{N}$, i.e. $P(\{i\})=\frac{1}{2^i}$. I know $P$ is defined on the power set of $\mathbb{N}$. But is it true that for every rational number $0\le\...
phst's user avatar
  • 586
0 votes
2 answers
31 views

Help with finding the number of terms in a geometric sum

I'm currently stuck on a problem involving a geometric sum, and I was hoping to get some assistance with it. Here's the problem: A geometric sum is equal to 215. The first term is 5, and the last term ...
Bishop_1's user avatar
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