robjohn

West Hills, CA

none

Age: 55

the mean square The Mean Square
(with one standard deviation and several unusual ones)

aka Rob Johnson

$\LaTeX$ support for Chat

12h
comment For which values of $x_0\in\mathbb{R}$ is $f$ differentiable
You can, but since $\frac1{x-1}$ is differentiable for $x\ne1$ and $(x-1)^3$ and $\sin(x)$ are differentiable everywhere, you can use the chain and product rules. That is, unless your instructions were to use only the definition of the derivative.
13h
comment For which values of $x_0\in\mathbb{R}$ is $f$ differentiable
@snowman: everything other than $x_0=1$ is gotten by simply using standard chain and product rules.
13h
comment For which values of $x_0\in\mathbb{R}$ is $f$ differentiable
@snowman: since $(x-1)^2\to0$ and $\left|\sin\left(\frac1{x-1}\right)\right|\le1$, the limit of their product is $0$. If you are still not convinced, use $$-(x-1)^2\le(x-1)^2\sin\left(\frac1{x-1}\right)\le(x-1)^2$$ and the Squeeze Theorem.
14h
revised Evaluate $\int_0^{\infty} \frac{\log x }{(x-1)\sqrt{x}}dx$ (solution verification)
add a link to the digamma function
15h
awarded Nice Answer
1d
answered Determining the value of an integral using complex methods
1d
comment Comments are not for extended discussion; this conversation has been moved to chat.
There will be a link in the comments to the chatroom used for the discussion so it can be referenced easily.
1d
revised Using Fourier analysis to show a function is positive
answer the second question and add some motivation
1d
revised Prove that $0^0 = 1$ using binomial theorem
add necessary comma
1d
answered Prove that $0^0 = 1$ using binomial theorem
1d
comment Using Fourier analysis to show a function is positive
@Fabian: Indeed. This is another way of proving that $$\sum_{n\in\mathbb{Z}}\frac1{n^2+1}=\pi\coth(\pi)$$
1d
answered Using Fourier analysis to show a function is positive
1d
comment Properties of carry in base $b$ multiplication
Carries happen when adding two numbers. Multiplication is repeated addition, so that might be what you are asking. Can you clarify what you are asking?
2d
revised Evaluate $\int_0^1 \frac{x^k-1}{\ln x}dx $ using high school techniques
word more appropriately
2d
revised Evaluate $\int_0^1 \frac{x^k-1}{\ln x}dx $ using high school techniques
shorten comment to $(8)$ to fit better
2d
revised Evaluate $\int_0^1 \frac{x^k-1}{\ln x}dx $ using high school techniques
simplify so as not to use convergence theorems
2d
answered Evaluate $\int_0^1 \frac{x^k-1}{\ln x}dx $ using high school techniques
2d
comment Converging Sequences
Interesting. Tangential, however, since the question says that $x_1\gt0$ and $a\gt0$. It might be an interesting project to find how the sequences behave for different $a\lt0$ and $x_1$. Since $x_{n+1}/x_n$ is even, the sign of $x_1$ doesn't provide anything interesting.
2d
comment Arranging set A and B to maximize their power
I imagine something like $(a_1b_1+a_2b_2)-(a_1b_2+a_2b_1)=(a_2-a_1)(b_2-b_1)$.
2d
comment Arranging set A and B to maximize their power
The rearrangement inequality is useful in many circumstances (+1)
1 2 3 4 5