## 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?