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Questions tagged [sequences-and-series]

for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.

for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.

1721 questions
15
votes
3 answers

Infinite series with signed sums

This was asked earlier at MSE. Let $A = \{a_0, a_1, a_2, \dotsc\}$ denote a weakly decreasing sequence of positive terms whose sum converges. Next introduce plus minus signs in every possible way, to form the collection $L(A) = \{\sum…
user2052
  • 1,401
14
votes
1 answer

Calculation of a series

It seems that we have: $$\sum_{n\geq 1} \frac{2^n}{3^{2^{n-1}}+1}=1.$$ Please, how can one prove it?
Aligator
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14
votes
3 answers

sequences of real numbers

Let $\lbrace x_i\rbrace_{i=1}^\infty$ be a sequence of distinct numbers in $(0,1)$. For any $n$ after deleting $x_1,...,x_n$ from $[0,1]$ we get $n+1$ subintervals. Let $a_n$ be the maximum length of these subintervals. Is there any sharp lower…
dimo
  • 191
12
votes
2 answers

Finding a closed form for $\sum_{k=1}^{\infty}\frac{1}{(2k)^5-(2k)^3}$

I'm looking for a closed form for the expression $$ \sum_{k=1}^{\infty}\frac{1}{(2k)^5-(2k)^3} $$ I know that Ramanujan gave the following closed form for a similar expression $$ \sum_{k=1}^{\infty}\frac{1}{(2k)^3-2k}= \ln(2)-\frac{1}{2} $$ I…
A.Neves
  • 536
10
votes
2 answers

fibonacci series mod a number

I'm trying to write a program with an input of numbers $n$ and $k$ (where $n<10^{1000}$ and $k<10^9$), where I compute fib[n] % k. What is a good FAST way of computing this? I realize that the resulting series is periodic, just not sure how to find…
user9734
  • 103
9
votes
1 answer

Generalized Vieta-product

It's known that $$S_2={2\over\pi} = {\sqrt{2}\over 2}{\sqrt{2+\sqrt{2}}\over 2}{\sqrt{2+\sqrt{2+\sqrt{2}}}\over 2}\dots$$ The terms in the product approaches 1, the same holds for the following convergent series, with $\phi$ the golden ratio …
robotic
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8
votes
2 answers

The sum of a series

Let $0< \alpha <1$ and $q>1.$ Consider the (alternating) series: $$ \sum_{k=1}^\infty (-1)^k \frac{q^k (q^k-1)^\alpha}{(q^k-1)\dots (q-1)}.$$ Denote its sum by $f(q,\alpha).$ Prove (or disprove) that $f(q,\alpha)\neq 0$ for…
Deepti
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8
votes
1 answer

Is this series well known?

I recently encountered the following function $$ f(t) = \sum_{n=0}^\infty \frac{t^{n^2}}{n^2!}. $$ It seems familiar, though I cannot remember where I might have seen it before. I would like to know in what text, if any, it has been studied. Above…
Graham Smith
  • 168
7
votes
3 answers

Series solution for general trinomial

Consider the equation $x^5-2x^2+z=0$ How do you derive the Lagrange inversion theorem series solution for it? I know it exists because the answer is here for any trinomial https://arxiv.org/pdf/0910.2957.pdf I am trying to figure out how derive the…
CarP24
  • 327
6
votes
1 answer

Asymptotic behaviour of a sequence

Hello, I am interested in some kind of sequence that are "not finitely recurrent". Let $a_i$ be a sequence taking values in $\{0,1\}$. Consider the sequence $(u_i)$ such that $u_0=1$, and for any positive integer $n$, $u_n =\sum_{i=1}^n…
Nekochan
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6
votes
1 answer

Convergence of derived series

There are quite a few simple results about convergent/divergent series derived from similar ones. So, here is something that I came across recently: Let $\sum_{n=1}^\infty a_n$ consist of positive terms and be convergent. Define $b_n =…
Ivan
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6
votes
0 answers

Tweak the numerators in the alternating harmonic series so that the partial sums alternate across $0$. What's the pattern in the numerators?

I was thinking about the alternating harmonic series: $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots$. I wondered what would happen if we tweak the numerators so that the partial sums alternate between "barely" non-negative and "barely"…
Dan
  • 2,431
6
votes
0 answers

Finding limit points of $\{2^n \sqrt 2\}$

How can I find limit points of $\{2^n \sqrt 2\}$, where $\{\cdot\}$ denotes the fractional part function? This is a subsequence of the sequence $\{\sqrt n\}$ for which we know the set of limit points. However, it is not clear to me if $\{2^n…
Rabat
  • 61
6
votes
6 answers

Fibonacci sequence inversion

How do I get the index in the sequence from the Fibonacci number? 0 1 1 2 3 5 8 13 21 34... For example N(3) = 4 (starting from zero) N(34) = 9 (starting from zero) .. N(X) = ? I've seen an equation in wikipedia There are other ways to compute it?
Hernán Eche
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6
votes
1 answer

On convergent series - in the spirit of Abel and Dini

Nonexistence of boundary between convergent and divergent series? I'm hoping the following is true: Suppose $a_i $ is a positive sequence and $\sum_i a_i < \infty.$ Then there exists a positive sequence $b_i$ s.t $\sum_i b_i < \infty$ and $\sum_i…
Better2BLucky
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