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 Yearling
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2d
answered Find the area bounded between $f(x)=\frac{\arctan(x)}{x^2}$ and $g(x)=\frac{\arctan(x)}{x^2+1}$
2d
comment Strange integral test for convergence in my Analysis Script (proof flawed ?)
The real point is that the integral test usually involves starting with some $f(x)$ such that $a_n=f(n)$. The claim you state works sort of the other way. You have a sequence $a_n$ from which you define what $f$ is. So that work is done ahead of time before the if and only if statement.
2d
comment Strange integral test for convergence in my Analysis Script (proof flawed ?)
The Theorem you state, which is basically correct, is not the same thing as the integral test you link to in Wikipedia.
2d
awarded  Yearling
Jun
24
comment $\sum a_n$ converges $\iff$ $\sum f(a_n)$ converges
Okay, got you..
Jun
24
comment $\sum a_n$ converges $\iff$ $\sum f(a_n)$ converges
We have nonzero terms. Strictly positive.
Jun
20
revised A proof that $EX_n\to EX$ for uniformly integrable $\{X_n\}$ with $X_n\to X$ a.s.
deleted 2 characters in body
Jun
20
comment A proof that $EX_n\to EX$ for uniformly integrable $\{X_n\}$ with $X_n\to X$ a.s.
Yes, the convergence is uniform on the special set $A_\delta$ by Egoroff's theorem. Re-read the answer more carefully. If you haven't heard of Egoroff's theorem this is probably little use.
Jun
20
answered A proof that $EX_n\to EX$ for uniformly integrable $\{X_n\}$ with $X_n\to X$ a.s.
Jun
14
comment Solving heat equation with Dirichlet boundary conditions on an interval
Solve both the equations $X_{xx}/X=-\lambda$ and $T_t/KT = -\lambda$. Then find which $\lambda$ allow for the boundary conditions to be met. Finally, use fourier series to keep the initial conditions.
Jun
9
comment How many possibilities of writing a natural number $M$ as a sum of $N$ natural numbers between $0$ and $M$?
The formula should be ${M+N \choose N}$, I believe.
Jun
9
revised When does $(x^x)^x=x^{(x^x)}$ in Real numbers?
added 145 characters in body
Jun
9
answered When does $(x^x)^x=x^{(x^x)}$ in Real numbers?
Jun
4
answered Is there an easy way to see that $E(X^2) \geq E^2(X)$?
Jun
1
comment Solutions of the equation $X^n-1\equiv 0$ (mod $m$)?
Isn't this something like the sum of all primitive roots across divisors?
May
29
answered $\gcd(a,n)\neq 1 \implies $ there is $b$ such that $ab\equiv 0 \pmod{n}$
May
25
comment If two different linear combinations of two random variables are Gaussian, can we deduct both of them are Gaussian.
What conditions are required? Linear independence? I realized I assumed that the random variables are linearly independent.
May
25
answered If two different linear combinations of two random variables are Gaussian, can we deduct both of them are Gaussian.
May
25
comment Convergence of complex power series $z^{n!}$ at boundary
The last step uses uniform convergence, right?
May
15
answered Region bounded by $x=y^2$ and $x=y^3$