All Questions
Tagged with sequences-and-series geometric-series
386
questions
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T/F: If $\alpha\in\mathbb{R}_{>1}$ is not Pisot, then $\{\lfloor\alpha^n\rfloor:n\in\mathbb{N}\}$ contains infinitely many odd and even integers
True or false:
If $\alpha\in\mathbb{R}_{>1}\setminus\mathbb{N},$ then
the set $\{ \lfloor \alpha^n \rfloor: n\in\mathbb{N} \}$ contains
infinitely many even integers and infinitely many odd ...
- 20.7k
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2
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73
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Telescopic summation for AGP: $R_n=\sum_{k=1}^n k r^{k-1}$
By the text-book method the summation of AGP is well known as: $$R_n=\sum_{k=1}^n k r^{k-1}=\frac{1-r^n-nr^n(1-r)}{(1-r)^2}.......(*)$$
We can get summation of a GP $(S_n=\sum_{k=1}^{n} r^{k-1})$ ...
- 43.6k
9
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Is there a non-trivial arithmetic progression of positive integers such that every number contains the digit $2?$
Let $X:=\{$ positive integers that contain the digit $2\}$
For fixed $m,n\in\mathbb{N},$ define the A.P. $S_{m,n:=}\ \{m,\ m+n,\ m+2n, \ldots\}\ .$
I am interested in $S_{m,n}\cap X,$ and $S_{m,n}\cap ...
- 20.7k
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40
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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 ...
- 49
1
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99
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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 ...
- 19
-2
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2
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96
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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 ...
- 135
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votes
1
answer
20
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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 ...
- 185
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1
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86
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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
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1
answer
44
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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 ...
- 176
1
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1
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80
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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 ...
- 20.7k
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1
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46
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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.
...
- 506
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1
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36
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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 ...
- 103
3
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114
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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 ...
- 133
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58
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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 $a_n=1/2^n$, ...
- 477
-1
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1
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43
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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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