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.

Thanks in advance

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?

Thanks in advance.

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 doesn’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?