## real analysis – Solve \$lim_{xto 0} frac {arcsin(x) sqrt{sin(x)}}{tg(x)}\$

This is an exam problem from my analysis 1 course, and I can’t find a way to solve it.

$$lim_{xto 0} frac {arcsin(x) sqrt{sin(x)}}{tgx}$$

So far I tryed applying L’Hopital’s rule, since as x approaches zero, we get $$frac{0}{0}$$ as a result, and eventually I got

$$lim_{xto 0} frac{frac{sqrt{sin(x)}}{sqrt{1-x^2}}+frac{arcsin(x)cos(x)}{2sqrt{}sin(x)}}{frac{1-x}{sqrt2x-x^2}}$$

Fast forward few steps, after trying to get rid of the double fractions, I got

$$lim_{xto 0}frac{sqrt{2x-x^2}(2sin(x)+sqrt{1-x^2}arcsin(x)cos(x))}{2(1-x)(sqrt{1-x^2})(sqrt{sinx})}$$

And I get again $$frac{0}{0}$$ when I let x approach zero. I honestly doubt I should apply L’Hopital again here, and since I can’t see any other way around I am asking for help or a clue how to solve this problem.

## inequality – If \$a^2 + b^2 + c^2 = 1\$, what is the the minimum value of \$frac {ab}{c} + frac {bc}{a} + frac {ca}{b}\$?

Suppose that $$a^2 + b^2 + c^2 = 1$$ for real positive numbers $$a$$, $$b$$, $$c$$. Find the minimum possible value of $$frac {ab}{c} + frac {bc}{a} + frac {ca}{b}$$.

So far I’ve got a minimum of $$sqrt {3}$$. Can anyone confirm this? However, I’ve been having trouble actually proofing that this is the lower bound. Typically, I’ve solved problems where I need to prove an inequality as true, but this problem is a bit different asking for the minimum of an inequality instead, and I’m not sure how to show that $$sqrt {3}$$ is the lower bound of it. Any ideas?

## How to show that if \$[x]= v \$ then \$ left | x-v right | < frac {1} {2} \$

How to show that if $$[x] = v$$ then:

$$left | x-v right | < frac {1} {2}$$

Or $$[]$$ is the closest whole function.

I can round a real number, but how do I prove it? It sounds simple, but I couldn't prove it.

## calculus – Calculate the Taylor series \$ frac {1} {x ^ 2 + 4x + 3} \$ to \$ x = 2 \$

I was trying to solve a declared manual exercise as follows:

Use the completion of the square and the geometric series to get the Taylor extension $$x = 2$$ of $$frac {1} {x ^ 2 + 4x + 3}$$

My first attempt was

$$frac {1} {x ^ 2 + 4x + 3} = -1 ( frac {1} {1- (x + 2) ^ 2})$$, even if the expression inside the parenthesis is in the form of a geometric series $$frac {1} {1 – x}$$. I realized that $$x = 2$$ is not in the convergence domain of geometric series, that is to say $$| (x + 2) ^ 2 | < 1$$. So I must be wrong in this direction.

Could you please provide me with some other directions to work with?

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

Denote
$$K_t (x) = frac {1} {(4 pi t) ^ { frac {n} {2}}} exp (- frac {| x | ^ 2} {4t})$$
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 $$L ^ p (p ge1)$$. Now how can we prove that $$frac {f (x) – (f * K_t) (x)} {t} to – Delta f (t to 0)$$
holds ($$Delta$$ designates the Laplace operator)? I have been given proof using spectral resolution, but I wonder if there is simpler proof using basic analysis. Thank you.

## solve \$ oint_ {| z | = 1} ^ {} frac {1} {z-2} dz \$

I need to solve $$oint_ {| z | = 1} ^ {} frac {1} {z-2} dz$$

I think I can use the residue theorem and therefore I get 1 for the residue and in the set $$2 pi i$$ I think?

But what do I do with $$| z | = 1$$?

## nt.number theory – Is there a non-negative sequence \$ a_p \$ such that \$ sum_p frac {a_p} {p} \$ converges but \$ sum_p frac { sqrt {a_p}} {p} \$ diverge ?

Is there a true non-negative sequence $$a_p$$ indexed on prime numbers such as $$sum_p frac {a_p} {p}$$ converges but $$sum_p frac { sqrt {a_p}} {p}$$ diverges? If so, what is an example of such a sequence, and if not, how can it be proven?

(This appeared when studying the pretension distance on multiplicative functions in analytic number theory. A sequence satisfying these conditions is necessary to find a multiplicative function $$f$$ such as $$sum_p frac {1 – Re (f (p))} {p}$$ converges but $$sum_p frac {| 1 – f (p) |} {p}$$ diverges.)

## Prove begin {equation} notag lim_ {n to infty} frac {1} {n} sum_ {i = 1} ^ {n} a_ {i} = a. end {equation}

I have to prove this lemma:

Let $${a_ {n} } _ {n = 1} ^ { infty}$$ a sequence of non-negative real numbers, and let $$lceil x rceil$$
the smallest integer $$n$$ such as $$x le n$$, or $$x$$ is a real number. If for each $$r> 1$$,
$$start {equation} notag lim_ {n to infty} frac {1} { lceil r ^ n rceil} sum_ {i = 1} ^ { lceil r ^ n rceil} a_ {i} = a, end {equation}$$
then
$$start {equation} notag lim_ {n to infty} frac {1} {n} sum_ {i = 1} ^ {n} a_ {i} = a. end {equation}$$

I know I have to use the limit definition (with epsilons ..) but I don't know how. Can you give me a hint?

## functions – Rewrite of \$ frac {d} {dx} min (x, n + 1) \$ using only basic arithmetic.

So I wonder if it is possible to find an expression equal to an expression: $$frac {d} {dx} min (x, n + 1)$$ which is defined only by basic operations? All integer variables, so it doesn't matter that as a function, they are discontinuous.

I was trying to find a formula for the sum of the squares with an arbitrary lower limit: $$sum_ {k = x} ^ n k ^ 2$$, but could not find $$sum_ {k = 1} ^ n k ^ 2 = frac {n (n + 1) (2n + 1)} {6}$$, which only works if the lower limit is 1 (or 0).

I discovered that you can express the first more general sum like:

$$sum_ {k = x} ^ nk ^ 2 = biggr ( frac {d} {dx} min (x, n + 1) biggr) biggr ( sum_ {k = 1} ^ {- x } k ^ 2 + sum_ {k = 1} ^ nk ^ 2 – sum_ {k = 1} ^ {x-1} k ^ 2 biggr)$$

It means I can express more general sum of squares using the sums of squares with a lower limit equal to 1. Therefore, I can rewrite the expression as:

$$sum_ {k = x} ^ nk ^ 2 = biggr ( frac {d} {dx} min (x, n + 1) biggr) biggr ( frac {n (n + 1) (2n +1)} {6} – frac {x (x-1) (2x-1)} {6} biggr)$$

I'm happy with the expression in the right parentheses, but it only works with the extraterrestrial expression in the left parentheses, so I'm trying to find an equal expression that doesnâ€™t use than basic operations.

Since English is not my native language, I may have used some terms incorrectly or clumsily. But I hope the question makes sense. Plus, I'm really not that good at math, so it's very likely that I made a mistake, or I make it unnecessarily complicated.

## nt.number theory – How I can prove or deny that \$ frac {x} {y + z} + frac {y} {x + z} + frac {z} {y + x} = 1 \$ in rational ?

The motivation for this question is to seek if there is such a rational number solution to the identity that is mentioned here, I have made many attempts using wolfram alpha to find such pairs of rational $$(x, y, z)$$ For who $$frac {x} {y + z} + frac {y} {x + z} + frac {z} {y + x} = 1$$ in rational but I failed even if I believed that there were no such solutions?