List of maths fields I have interest in:
1. Group theory in terms of orbits and actions
2. Zero term algebra and division by zero algebra
3. Integration in the language of abstract algebra and as a functional, symmetry of integrands
4. Optimising proofs given axiomatic systems
5. Category theory
6. Unnatural algebraic structures
7. Patterns in expanding multiplications of polynomials
8. Set of all counterexamples given a proposition
9. Tensor visualisation and intuitions
10. Numerical analysis methods to explore special regions or points of mathematical functions or systems of equations

$$I=\int x^2 \sin x e^x dx$$
Using Feymann's integration trick (Special case of Lebniz rule) and $\sin x= \mathcal{I}(e^{ix})$
\begin{align}
I & =\left.\frac{d^2}{dt^2}\int -\sin (tx) e^x dx\right|_{t=1}\\
& = \left.\frac{d^2}{dt^2}\int \mathcal{I}(-e^{itx}) e^x dx\right|_{t=1}\\
& = \left.-\frac{d^2}{dt^2}\mathcal{I}\left(\int e^{(it+1)x} dx\right|_{t=1}\right)\\
& = \left.-\frac{d^2}{dt^2}\mathcal{I}\left(\frac{1}{it+1}e^{(it+1)x}\right|_{t=1}\right)\\
& = \left.-\mathcal{I}\left(\left(\frac{-2}{(it+1)^3}+\frac{-2i}{it+1}+(it+1)\right)e^{(it+1)x}\right|_{t=1}\right)\\

That is one of simpleart's challenges. While I am well aware that divergent series have no well defined values (and I am **not** going to compute the so called 'value' in the fashion of 1+2+3+4... because that is nonrigorous cheating.)

The challenge has a hint that somehow, some notion of manipulating it is actually rigorous and I suspect we are perhaps calculating some kind of limit of an infinite polynomial with coefficients from that series such that setting the polynomial at $x\to 1$ will arbitrarily gave a limiting value that the series should be, analogous to computing limit of func…

I am looking at the proof of the following theorem:

Let $E/F$ be a cyclic extension of degree $n$ and $F$ contains a $n$-th primitive root. Then $\exists a\in F$ such that $E$ is the splitting field of $f=x^n-a$ and $f$ is irreducible over $F$.

The proof is the following:

We have that $\text{Gal}(E/F)=\langle \sigma \rangle$, $\text{ord}(\sigma)=|\text{Gal}(E/F)|=[E:F]=n$.

So, the $id, \sigma , \ldots , \sigma^{n-1}$ are different.

From a theorem we have that $\exists \beta\in E$ : $$\theta:=\beta+\omega\sigma (\beta)+\ldots +\omega^{n-2}\sigma^{n-2}+\omega^{n-1}\sigma^{n-1}(\beta)\neq…

For the middle step, if we apply p series test to all the kth partial sums, it should converge absolutely, thus the limit will converge to some well defined value

Evaluating however is a different story (which is why I suck at series, especially this one has no factorials to allow me to pattern match some exp or trigonometric functions)

DHMO: If that's the closed form of the given series, then its limit will be 1/2, but... isn't 1/(x+1)^2 continuous at x=1 thus it is not really representing the neighbourhood of x=1 for the power series p(x)?