Questions tagged [sequences-and-series]
For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.
67,756 questions
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Encoding prefixes of infinite sequences.
We consider infinite binary sequences $x \in[0,1]$ via their binary expansion.
$O: \{0,1\}^* \rightarrow \{0,1\}^*$ maps finite binary strings to finite binary strings.
for a string $m$ we define:
$$
...
- 61
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Is this a proper way of looking at sorted lists of numbers?
Suppose we have a list L[i] of numbers, of length n. It seems appropriate to say that L is given in ascending order if i < j implies L[i] <= L[j].
On the other hand, it seems sufficient to show ...
- 1,262
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Prime-polydivisible numbers: proving the maximum length without computer search
In a book I read about polydivisible numbers in base $10$.
A polydivisible number is a number where the number formed by its first $n$ digits is divisible by $n$.
The largest of these numbers has $...
- 2,836
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Density of positive numbers in the sequence defined by $f_{n} = 1-f_{n-1}/f_{n-2}$
Found this interesting recursive sequence but im having a hard time proving this conjecture. Any help would be much appreciated!
Let $(f_n)_{n \ge -1}$ be defined by
$$
f_{-1} = -1, \quad f_0 = 1, \...
- 81
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Proof of hypergeometric transformation of well-poised series
so i want help in the proof of this transformation
$$
{}_{s+4}F_{s+3} \left[ \begin{matrix} a, b, c, a_1, a_2, \dots, a_s, -m; \\ 1+a-b, 1+a-c, p_1, p_2, \dots, p_s, p_{s+1} \end{matrix} \right] \\
= \...
- 431
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Does $f(x) = \sum_{k=1}^{\infty} (-1)^{k+1} \sin(x/k)$ have infinitely many nonreal zeros?
Consider the function
$$f(z) = \sum_{k=1}^{\infty} (-1)^{k+1} \sin\left(\frac{z}{k}\right)$$
I wonder where the nonreal zero's are of this function.
$$f(z) = \sum_{k=1}^{\infty} (-1)^{k+1} \sin\left(\...
- 19.4k
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Convergence of the infinite product $\prod_{n=1}^{\infty} \left( 1+\frac{x^{n}}{n^{p}} \right)\cos \left( \frac{x^{n}}{n^{q}} \right)$ for $|x| > 1$
I encountered the following problem in a mathematical analysis problem book (Demidovich). Let $p > 0$ and $q > 0$. We need to discuss the convergence of the following infinite product:
$$\prod_{...
- 51
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"Dominance" theorem for convergence of series
UPDATE: both definitions for "dominance" given in the post have counterexamples. If there is some "dominance theorem" that exists it would have to use a different definition than I ...
- 71
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If $f(p)=\left(\sum_{x=1}^{\infty}\frac{x^2}{(x+p)(x+2p)(x+3p)(x+4p)}\right)^{-1}$ then find value of $30\times \frac{f(7)}{f(3)}$
Let $f(p)$ be defined for any natural number $p$ as $$f(p)=\left(\sum_{x=1}^{\infty}\frac{x^2}{(x+p)(x+2p)(x+3p)(x+4p)}\right)^{-1}$$ then find value of $$30\times \frac{f(7)}{f(3)}$$.
By using method ...
- 11.6k
4
votes
4
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$n$th Element of the Connell Sequence $1, 2, 4, 5, 7, 9, 10, 12, 14, 16, \dots$
The sequence can be understood as following,
$$\{1\}, \{2, 4\}, \{5, 7, 9\}, \{10, 12, 14, 16\}, \dots$$
Below is the definition from MathWorld
The Connell sequence is the sequence obtained by ...
- 648
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4
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Query on the definition of convergence of a sequence
I'm working my way through an intro Numerical Analysis textbook and I've come across a definition on limit for an infinite sequence of real numbers (see image attached).
I was hoping to obtain some ...
- 438
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1
answer
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Confused about $S=\sum _{r=1}^{\infty} \left(\frac{2}{4r-3}-\frac{1}{2r}\right)$
It is regarding the sum
$$S=\sum _{r=1}^{\infty} \left(\frac{2}{4r-3}-\frac{1}{2r}\right)$$
in MSE:
A Question based on series.
Where the answer is given as
$$S=\frac{\pi}{4}+\frac{3}{2} \ln2,\tag{*}$$...
- 873
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Expand an elementary function into a Maclaurin series
In general, if
$$
\ln f(x)=\sum_{k=0}^\infty a_k x^k,
$$
what is the Maclaurin series expansion of the function
$$
[f(x)]^\lambda=\exp\Biggl(\lambda\sum_{k=0}^\infty a_k x^k\Biggr)
$$
around $x=0$?
...
- 2,362
1
vote
2
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Sequence Problem: How can the evidence for the missing part in the proof be found?
Consider an array $h$ such that its elements are defined as follows:
$$\begin{eqnarray*} h_0 & = & 2 \text{,} \\ h_1 & = & 3 \text{,} \\ h_2 & = & 6 \text{,} \\ h_n & = &...
- 1,730
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Trying to evaluate $\sum_{n = 0}^\infty \frac{1}{n^2+1}$ using a Laplace Transform.
Recently I came across the infinite sum $\sum_{n = 0}^\infty \frac{1}{n^2+1}$ on Maths 505's wonderful YouTube channel. The solution I saw involved the digamma function, but looking at the series I ...
