Questions tagged [divergent-series]
The divergent-series tag has no usage guidance.
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A problem about the series $\sin(n^p)$ [closed]
Prove that when $p>0,$ the series $$\sum_{n=1}^\infty \sin(n^p)$$
is divergent
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How are distributions and divergent series summations related?
When we do Fourier analysis, we don't always get convergent series. A classic example comes from considering the Sawtooth function. It has Fourier Coefficients
$$s(x) = \frac{1}{2} + \sum_{n \neq 0} \...
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Does summing divergent series using cutoff functions give consistent results?
One way to try to give a value $S$ to a divergent series $\sum_{n=1}^\infty a_n$ is with a smooth cutoff function:
$$
S = \lim_{N\to\infty}\sum_{n=1}^\infty a_n \eta\left(\frac{n}{N}\right)
$$
where $\...
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Series convergence if $\sum a_n^2 < \infty$
There are quite a few simple results about convergent/divergent series derived from similar ones. Here is a question in the same spirit that I saw posted on another forum. Unfortunately, I don't have ...
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History of asymptotic expansion of Laplace’s method between Laplace and Erdélyi
In 1774 Laplace understood that $I≔∫\textrm{d}x \exp kf(x)$ for $k≫0$ can be estimated if one knows 2-jet of $f$ at its point of maximum (as $I₀ ≔ ∫\textrm{d}x \exp kf₀(x)$ with $f₀$ quadratic with ...
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Can we just use the linear term of exponential sums to sum divergent series
Suppose you want to compute the sum $\sum_{n=0}^{\infty} a_n $
You could consider the expression $f(x) = \sum_{n=0}^{\infty} e^{a_n x}$ and try to compute the coefficient of an $x^1$ term in the ...
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Laurent Series $\sum_{n=-1}^\infty a_n x^n$ when $a_{-1} = \infty$
When dealing with complex functions, if $f(x)$ has a simple pole, then we can find the coefficient $a_{-1}$ in the Laurent expansion $f(x) = \sum_{n=-1}^\infty a_n x^n$ by evaluating the limit $\lim_{...
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If $f(x) = \sum_{n=0}^\infty a_n x^n$, then $\int_{-\infty}^\infty f(x^2) dx = \pi i a_{-\frac{1}{2}}$
I've noticed a curious relationship between the coefficient $a_n$ for a power series and the integral of the real line. For instance, take $f(x) = e^{-x} = \sum_{n=0}^\infty \frac{(-1)^n}{n!} x^n$. ...
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Computing residues at $\infty$
As an initial note, let me show by example what I mean by the terminology 'residue at $\infty$' I use in the title. I assume there is some standard terminology for this stuff, so I'd appreciate it if ...
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Correct way to extend a sequence defined on the naturals into the complex plane
Preamble
Sequences $a_n$ defined on the natural numbers are clearly not uniquely interpolated by only one function. In particular, given an interpolation $f(n) = a_n$, then $f(n) + \sin(2\pi n)$ is ...
- 1,692
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$\frac {f (0)}{2}+ \sum_{k=1}^{\infty}f (k)=\sum_{n=-\infty}^{\infty} \mathcal{L} \{ f \} (2 \pi i n)$
I obtained the very strange formula above and at begining I was just wanted know how to interpretate it. But now when I know what is this with help of @Carlo Beenakker, I am leaving it as a proof. BTW ...
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Do roots of these polynomials approach the negative of the Euler-Mascheroni constant?
Let $p_n$ be the $n$th degree polynomial that sends $\frac{k(k-1)}{2}$ to $\frac{k(k+1)}{2}$ for $k=1,2,...,n+1$. E.g., $p_2(x) = (6+13x -x^2)/6$ is the unique quadratic polynomial $p(x)$ satisfying $...
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1+2+3+4+… and −⅛
Is there some deeper meaning to the following derivation (or rather one-parameter family of derivations) associating the divergent series $1+2+3+4+…$ with the value $-\frac 1 8$ (as opposed to the ...
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Convergence and roots of alternating periodic infinite series
Let $0<\alpha <1$ and $\beta > 0$. Consider the mapping $$F(\alpha, \beta) = \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}.$$ Can we prove $F(...
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Fractional integrals and $\sum f(n) n^x$
Preamble
The following is a rather unrigorous way to obtain the Euler-Maclaurin formula. Consider some $\sum_{n=1}^\infty f(n)$. We may rewrite this as
$$\sum_{n=1}^\infty f(n)=\sum_{n=1}^\infty \sum_{...
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