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.
67,349 questions
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Evaluate $\int_{-\infty}^{+\infty} \frac{dx}{(\cos(x)+2)(1+x^2)}$ as an infinite series using residues
I am attempting a problem out of Weinberger's PDEs book, that is to solve the following integral
$$\int_{-\infty}^{+\infty} \frac{dx}{(\cos(x)+2)(1+x^2)}$$
as an infinite series via the residue ...
- 145
-4
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0
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Why Does This Sequence of Valid Strings Seem to Approach e? [closed]
I listed the number of all cases of a string consisting of: (-,S,c,P)
+Additional comment: There is no meaning to these characters, you can see them as ABCD.
like this:
...
- 9
3
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1
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87
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How to prove a limit relating strong divisibility sequences equals special values of the Riemann zeta function?
Recently, I came across the following result:
Let $F_n$ denote the $n$-th Fibonacci number, then $$\lim_{n\to\infty}\frac{\log(F_1\cdot\ldots\cdot F_n)}{\log(\mathrm{lcm}(F_1,\,\ldots,\,F_n))}=\zeta(2)...
- 6,243
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0
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Do unbounded sequences on lattices have subsequences for which no further subsequence is bounded?
Let $\Omega$ be a partially ordered set with partial order $\le$ and, furthermore, be an unbounded lattice (there exists a a smallest $x \vee y$ such that $x \le x \vee y$ and $y \le x \vee y$, and $x ...
- 1,509
3
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2
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A better method to derive a closed-form expression for the improper integral $\int_0^\infty \sqrt{t^2+1} \, e^{-\lambda t} J_0(\beta t) \, d t $
I would like to obtain a closed-form expression for the following improper integral:
$$
I = \int_0^\infty \sqrt{t^2+1} \, e^{-\lambda t} J_0(\beta t) \, \mathrm{d} t \, ,
$$
where the parameters $\...
- 486
5
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2
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Prove $\sum_{n=0}^\infty\frac{1}{n+z}\prod_{k=1}^n\frac{k}{k+z}=2z\sum_{n=0}^\infty\frac{(-1)^n}{(n+z)^2}$
Prove for $\Re(z)>0$ the identity $$\sum_{n=0}^\infty\frac{1}{n+z}\prod_{k=1}^n\frac{k}{k+z}=2z\sum_{n=0}^\infty\frac{(-1)^n}{(n+z)^2}$$ (with $\prod_{k=1}^n$ interpreted as $1$ at $n=0$).
This is ...
- 43.7k
4
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2
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87
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What concise method can solve the ODE, $y'' + \left(\frac{1}{x}+2x \right)y' + 4y =0$?
While solving the vorticity transport equation in cylindrical coordinates, $\frac{\partial \omega}{\partial t}=\nu \left(\frac{\partial^2 \omega}{\partial r^2}+\frac{1}{r}\frac{\partial \omega}{\...
2
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1
answer
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Infinite sums of nested radicals with closed-form expressions involving π [closed]
I found a quite striking formula that I would like to share, and I am interested in whether it has already been discovered. I fear so, as my methods aren't quite creative or original, but I would like ...
- 29
1
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1
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154
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Finding $\lim_{n \to \infty} \sum_{k=n}^{5n} \binom{k-1}{n-1}(1-x)^nx^{\,k-n}$
Recently I came across a problem that has me stumped. It is as follows:
Find the limit
$$
\lim_{n \to \infty} \sum_{k=n}^{5n} \binom{k-1}{n-1}(1-x)^nx^{\,k-n},\,\, x \in [0,1].
$$
I spent a long ...
- 129
0
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0
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Can I indicate that only even index values should be added in a summation like this? [duplicate]
Warning: I don’t know how to use MathJax very well but I can explain what I mean pretty easily.
If I have the summation symbol $Σ$ and I want to indicate that the summation should only include even ...
- 221
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0
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54
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Solving an exponential equation with arithmetic sequences
In a self made math problem I attempted to find a general formula for an arithmetic sequence $a_n=a_1+d(n-1) $ which satisfies that the sum of any $n$ terms is $n^k$ given a certain $k$ (ex: for $k=2$ ...
- 167
3
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Eigenvalues of variation of the Kac matrix
I am investiganting some regularity properties for vector fields on $SU(2)$ and I reduced it to obtaining lower bounds on the absolute value of the eigenvalues (which coincide with the singular values)...
- 1,443
9
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1
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+50
How to prove $\limsup_x\sum_{k\ge 0}\sin(2\pi x/2^k)/\log_2x=\sup_{n,k}\frac{1}{n}\sum_{j=0}^{n-1}\sin(\frac{2\pi k}{2^n-1}2^j)=\frac{\sqrt{15}}{8}$?
I am investigating the asymptotic behavior of the function
$$
g(x) = \sum_{k=0}^{\infty} \sin\left(\frac{2\pi x}{2^k}\right)
$$
and, in particular, seeking to determine the value of
$$\limsup_{x \to \...
- 1,407
0
votes
1
answer
49
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Is $ \sum_{n=0}^\infty \sum_{k=0}^\infty f(k, n + pk)=\sum_{n=0}^\infty\sum_{k=0}^{\left\lfloor \frac{n}{p} \right\rfloor} f(k, n)$Valid for Any $p$?
I tried to find a transformation property for a double summation. I discovered the following identity:
$$
\sum_{n=0}^\infty \sum_{k=0}^\infty f(k, n + 2k) = \sum_{n=0}^\infty \sum_{k=0}^{\left\lfloor \...
- 2,612
3
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0
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165
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+50
Can $\sum_{k=1}^{\infty}k \operatorname{sgn}(\cos(k))$ be normalized?
In this question, I considered a double-sequence $(x_k,y_k)$ whose first term was shown to be $x_k= k\operatorname{sgn}(\cos(k))$. I was interested when the quantity $X_n=\sum_{k=1}^n x_k$ or a ...
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