defined integrals – Prove $ int limits ^ { infty} _ {- infty} frac { sin (2n arctan (x))} { left (x ^ 2 + 1 right) ^ n left (e ^ {x pi} +1 right)} dx = eta (2n) – zeta (2n) $

For some time now, I have had to prove the following:

$$ int limits ^ { infty} _ {- infty} frac { sin (2n arctan (x))} { left (x ^ 2 + 1 right) ^ n left (e ^ {x pi} +1 right)} dx = eta (2n) – zeta (2n), $$

or $ eta (z) $ is the Dirichlet Eta function and $ zeta (z) $ is the Riemann Zeta function. I would like to prove it for $ n> $ 1, but showing that this is only true for an integer $ n $ would be nice too. For the whole case $ n $, I found an extension for the numerator which may be useful.

$$ sin (2n arctan (x)) = frac {x} {(x ^ 2 + 1) ^ n} sum ^ { left lfloor frac {2n-1} {2} right rfloor} _ {k = 0} left ( begin {matrix} 2n \ 2k + 1 end {matrix} right) (- x ^ 2) ^ k. $$

Other than that, I really couldn't do anything with this integral.

Find convex $ f $: $ limsup_ {n to infty}, a_n = left | frac { frac {d ^ {n + 1}} {dx ^ {n + 1}} e ^ {f (x)}} { frac {d ^ {n}} {dx ^ {n}} e ^ {f (x)}} right | _ {x = 1} $

Is there a convex function $ f $ that satisfies the following property: let
begin {align}
a_n = left | frac { frac {d ^ {n + 1}} {dx ^ {n + 1}} e ^ {f (x)}} { frac {d ^ {n}} {dx ^ {n}} e ^ {f (x)}} right | _ {x = 1}
end {align}

such as
begin {align}
limsup_ {n to infty} frac {a_n} {n} = infty
end {align}

But
begin {align}
liminf_ {n to infty} frac {a_n} {n} <1.
end {align}

real analysis – Suppose that $ limx_ {n} = x $ and $ x neq0 $, show that there is a $ N in mathbb N $ so that if $ n geq N $ then $ | x_ {n} | geq frac {| x |} {2} $

Yes {$ x_ {n} $} is a sequence of real numbers for which $ limx_ {n} = x $, and if $ x neq0 $, then prove that there is a $ N in mathbb N $ so if $ n geq N $ then $ | x_ {n} | geq frac {| x |} {2} $. Tip: use the positive number $ epsilon = frac {| x |} {2} $. You may need to use the inequality of the triangle to make a difference.

So far, I have: $ epsilon> 0 $ be arbitrary. Since $ limx_ {n} = x $, applying the definition of a limit $ epsilon = frac {| x |} {2} $, we have a $ N in mathbb {N} $ such as $ | x_ {n} -x | < frac {| x |} {2} $ for all $ n geq N $.

So I know that as long as I can prove that $ | x_ {n} |> frac {| x |} {2} $ then that is enough. However. I don't know where to use the given index of the inequality of the triangle. Any help is appreciated. Please also explain each step after what I have.

data structures – Proof that an almost complete binary tree with n nodes has at least $ frac {n} {2} $ leaf nodes

I'm having trouble proving what my title says. Some textbooks refer to almost complete binary trees as complete, so to be clear, when I say almost complete binary tree, I mean a binary tree whose levels are all full except the last one in which all the nodes are as far as possible on the left. .

I thought about proving it by induction but I'm not sure how to do it. Ideas?

convergence divergence – For which values ​​of p the improper integral $ int_0 ^ 1 frac {dx} {x ^ plnx} $ converges?

I'm having trouble finding the p values ​​for which the integral $ int_0 ^ 1 frac {dx} {x ^ plnx} $ converges.
So far, I've separated the integral into two separate integrals, $ int_0 ^ frac {1} {2} frac {dx} {x ^ plnx} $ and $ int_ frac {1} {2} ^ 1 frac {dx} {x ^ plnx} $, for the first one I found that it converges for values ​​of p where $ p gt1 $, but I don't seem to find a way to prove the convergence of the second integral, tried to use the power series for $ ln (x) $ but I'm not sure how to proceed, any information would be appreciated,
Thank you.

functional analysis – prove that $ int ^ 1_0 u ^ 2 dx le frac {1} {2} int ^ 1_0 (u & # 39;) ^ 2 dx ; forall u in B $

Let $ u: mathbb {R} à mathbb {R} $ the following takes $$ u (x) = u (y) + int ^ x_y u & # 39; (t) dt $$

let $ C & # 39; (0,1) $ be the set of functions $ u: (0,1) to mathbb {R} $ which are continuous and have on $ (0.1) $ consider the following subset of $ C & # 39; (0,1) $
$ B = {u in C & # 39; (0,1): u (0) = 0 } $

then prove that

$ int ^ 1_0 u ^ 2 dx le frac {1} {2} int ^ 1_0 (u & # 39;) ^ 2 dx ; forall u in B $$

how to start this question can anyone help me

sequences and series – Show that an explicit formula for $ u_r $ is given by $ u_r = 1+ frac {10} {3} [4^{r-1} -1]$

A sequence $ u_1, u_2, u_3 $, … is such that $ u_1 = $ 1 and $ u_ {n + 1} = 4u_n + 7 $ for $ n geqslant 1 $.

Note the first four terms in the sequence.

I solved the first half of the question.

$ T_1 = $ 1

$ T_2 = $ 11

$ T_3 = $ 51

$ T_4 = 211 $

What kind of sequence is this? It cannot be a geometric progression because there is no common relationship, nor can it be an arithmetic progression since 39; there is no common difference.

I need help solving the second half of the question.

Show that an explicit formula for $ u_r $ is given by $ u_r = 1+ frac {10} {3} [4 ^ {r-1} -1] $

How to show it? Should I use the formulas given in the question? Where is it $ u_r = S_r – S_ {r-1} $?

real analysis – Let $ X, Y $ be independent random variables normally distributed. Find the density of $ frac {X ^ {2}} {Y ^ {2} + X ^ {2}} $

Let $ X, Y $ be independent normalized normalized random variables and $ X, Y neq 0 $. Find the density of $ frac {X ^ {2}} {Y ^ {2} + X ^ {2}} $

I was given the tip of the first density calculation of $ (X ^ {2}, Y ^ {2}) $ then calculate the density of $ ( frac {X ^ {2}} {Y ^ {2} + X ^ {2}}, Y ^ {2} + X ^ {2}) $

When I follow the advice: I know that $ X ^ {2} $~$ Gamma ( frac {1} {2}, frac {1} {2}) $ and $ Y $ also. In addition, $ X ^ {2} $ and $ Y ^ {2} $ are still independent. Therefore, the density $ f _ {(X ^ {2}, Y ^ {2})} (x, y) $ can write $ f_ {X ^ {2}} (x) f_ {Y ^ {2}} (y) $ or $ f_ {X ^ {2}} $ and $ f_ {Y ^ {2}} $ are the density functions of $ X ^ {2} $ and $ Y ^ {2} $

My next idea, with the above advice in mind is to consider a card $ varphi: (x, y) mapsto ( frac {x} {x + y}, x + y) $

It ensues while $ ( frac {X ^ {2}} {Y ^ {2} + X ^ {2}}, Y ^ {2} + X ^ {2}) = varphi (X ^ {2}, Y ^ { 2}) $
and $ f _ { frac {X ^ 2} {Y ^ {2} + X ^ {2}}, Y ^ {2} + X ^ {2}} (a, b) = f _ { varphi (X ^ {2}, Y ^ {2})} (a, b)
$

Note that $ varphi ^ {- 1}: (a, b) mapsto (ba, b-ba) $ And so $ det D varphi ^ {- 1} (a, b) = det begin {pmatrix} b & a \
-b and 1-a
end {pmatrix} = b (1-a) + ab implies det D varphi ^ {- 1} ( frac {X ^ {2}} {Y ^ {2} + X ^ {2}}, Y ^ {2} + X ^ {2}) = (Y ^ {2} + X ^ {2}) (1- frac {X ^ {2}} {Y ^ {2} + X ^ {2} }) + ( frac {X ^ {2}} {Y ^ {2} + X ^ {2}}) (Y ^ {2} + X ^ {2}) = Y ^ {2} + X ^ { 2} $

And so $ P _ {( frac {X ^ {2}} {Y ^ {2} + X ^ {2}}, Y ^ {2} + X ^ {2})} (A) = int_ {A} f_ {( frac {X ^ {2}} {Y ^ {2} + X ^ {2}}, Y ^ {2} + X ^ {2})} (x, y) dxdy = int _ { varphi ^ {- 1} (A)} f _ {(X, Y)} (x, y) times (X ^ {2} + Y ^ {²}) dxdy = int _ { varphi ^ { – 1} (A)} f_ {X} (x) times f_ {Y} (y) times (X ^ {2} + Y ^ {²}) dxdy $

Where should I go from here?

nt.number theory – Asymptotic behavior of $ sum_ {k = 1} ^ {n} frac {p_ {k + 1}} {p_ {k + 1} -p_k} $

I am referring to my previous question Asymptotic behavior of a certain sum of consecutive prime number relationships.
We can divide the sum
$$ sum_ {k = 1} ^ {n} frac {p_ {k + 1} + p_k} {p_ {k + 1} -p_k} $$
or $ p_k $ represents the first clue $ k $, in the next two

$ sum_ {k = 1} ^ {n} frac {p_ {k + 1}} {p_ {k + 1} – , p_k} $ ~ $ frac {n , (n + 1)} {e} , log log n $

$ sum_ {k = 1} ^ {n} frac {p_ {k}} {p_ {k + 1} – , p_k} $ ~ $ frac {(n-1) , n} {e} , log log n $

Is there anyone who can confirm this asymptotic behavior and, if it is correct, give a sketch of evidence?

computation – Values ​​of $ left (1+ frac {1} {n} right) ^ n $ on a calculator

I was reading a calculation book, and at one point, the following standard limit was mentioned:

$$ lim_ {n to infty} left (1+ frac {1} {n} right) ^ n = e $$

Then the author of this book invited the reader to try to calculate, using a calculator, the value of the expression

$$ left (1+ frac {1} {n} right) ^ n $$

for various inputs. He then said the results would look like this:

The author then invites the reader to try to understand why this is happening.

I thought it was an interesting question and one that actually shows how calculus can make concepts work that would not normally be quantifiable or measurable in a physical world.

My theory is that since any calculator has a certain precision, it can store values ​​up to, after a certain point, the expression
$$ frac {1} {n} $$ just gets rounded up to $ 0 $, are there not enough bits to store the floating point value.

The expression is then evaluated as $ 1 ^ n $ which is constantly equal to $ 1 $. It doesn't matter if ultimately the calculator can't track the value of the exponent and rounds it up to something else, because the value of the power will be $ 1 $ whatever the exhibitor ends up being.

Other theories?