## $int_0^{infty} frac {x.dx}{(x^2+a^2)^{frac 32}(x^2+b^2)}$

I am solving a problem where I need to find the charge distribution on a conducting plate, and the field due to it. I’m stuck on this integral.
$$int_0^{infty} frac {x.dx}{(x^2+a^2)^{frac 32}(x^2+b^2)}$$

## probability – Calculating the quotient $frac { mu (E)} { nu (E)}$ of two measures with density w.r.t. the Lebesgue measure

Suppose that $$mu$$ and $$nu$$ are probability measures on $$mathbb {R}$$ with density $$f$$ and $$g$$ w.r.t. the Lebesgue measure, i.e. $$mu = fdx$$ and $$nu = g dx$$. Is there an easy way to calculate the qutotient
$$frac { mu (E)} { nu (E)}$$
of a measurable set $$E$$? This is a quotient of two integrals of $$f$$ and $$g$$. Can we calculate one single integral over the quotient of $$f$$ and $$g$$ instead?

## What do I use for $sum_ {n = 1} ^ { infty} (-1) ^ {n} frac { cos (n)} {n sqrt {n}}$

I think this is a comparison test problem, but I don't know what to compare it with. Any help would be appreciated, thanks. (Attempt to prove convergence).

## Is there a measurable function f: $mathrm forall a int_ {0} ^ {1} { frac {1} {f (x) -a}} , d mu < infty$?

I want to find a measurable function f: $$mathrm forall a int_ {0} ^ {1} { frac {1} {f (x) -a}} , d mu < infty$$, but I can't find such a function. However, I cannot prove that there is no such function. Could someone help me?

## Limits – Prove that $frac {f (x) – (f * K_t) (x)} {t} bis – Delta f (t bis 0)$ for $f in C_0 ^ { infty} ( mathbb {R} ^ n)$

Describe
$$K_t (x) = frac {1} {(4 pi t) ^ { frac {n} {2}}} exp (- frac {| x | ^ 2} {4t})$$
the heat core, $$f in C_0 ^ { infty} ( mathbb {R} ^ n)$$. It is known that $$f * K_t to f (t to 0)$$ point by point and $$L ^ p (p ge1)$$. How can we prove it? $$frac {f (x) – (f * K_t) (x)} {t} bis – Delta f (t bis 0)$$
holds ($$Delta$$ denotes the Laplace operator)? I got a proof with the spectral resolution, but I wonder if there is a simpler proof with the basic analysis. Thanks a lot.

## ordinary differential equations – what is the singular point of the ode $frac {dy} {dx} = sqrt {y} + 1$

But avoid

• Make statements based on opinions; Support them with references or personal experiences.

Use MathJax to format equations. MathJax reference.

## real analysis – why $sum_ {n = 1} ^ infty frac {4 sin pi n} { pi pi n ^ 2} sin nx = 2 sin (x)$?

I used Wolfram Alpha to calculate this integral
$$frac {1} { pi} int _ {- pi} ^ { pi} 2 sin x sin nx ; dx = frac {4 sin pi n} { pi pi n ^ 2}$$

So if $$n in mathbb {Z}$$ That answer must be $$0$$ Law?

Then I used Wolfram Alpha again for the calculation
$$sum_ {n = 1} ^ infty frac {4 sin pi n} { pi pi n ^ 2} sin nx$$
and it is called
$$sum_ {n = 1} ^ infty frac {4 sin pi n} { pi pi n ^ 2} sin nx = 2 sin (x)$$
How is that possible?

## $Frac {1} {6-n ^ 3}$ applies to all $n> 1$. Is it convergent or divergent?

So we're using the threshold comparison test, and we've got that $$lim_n rightarrow infty frac {n} {6-n ^ 3}$$. I know it's all over $$sum$$ from $$n = 1$$ to $$infty$$ from $$a_b$$, but I do not know what $$a_b$$ is.

## Algebra precalculation – Find $n$ such that $365 left (1 – ( frac {364} {365}) ^ n – n frac {364 ^ {n-1}} {365 ^ n} right)>$ 1

i have to find $$n$$ so that $$365 left (1 – ( frac {364} {365}) ^ n – n frac {364 ^ {n-1}} {365 ^ n} right)> 1$$. The answer is $$n ge 28$$. I understand expanding the equation, rearranging it, taking the logarithm
$$log (364)> log (365) + n log (364/365) + log (n) + (n-1) log (364/365).$$

I don't know what to do next. I would be happy if you could give me a hint.

## nt.number theory – is there a non-negative sequence $a_p$ so that $sum_p frac {a_p} {p}$ converges but $sum_p frac { sqrt {a_p}} {p}$ diverges?

Is there a real, non-negative sequence? $$a_p$$ indexed on the prime numbers so that $$sum_p frac {a_p} {p}$$ converges however $$sum_p frac { sqrt {a_p}} {p}$$ diverged? If so, what is an example of such a sequence, and if not, how can this be proven?

(This was the result of examining the presumption distance for multiplicative functions in analytical number theory. A sequence that meets these conditions is required to find a multiplicative function $$f$$ so that $$sum_p frac {1 – Re (f (p))} {p}$$ but converges $$sum_p frac {| 1 – f (p) |} {p}$$ diverges.)