## Prove $x_{n+1} = x_n + frac{x-x_n^2}{2}, x_0 = 0$ is increasing and less than $sqrt {x}$

Let $$x_{n+1} = x_n + frac{x-x_n^2}{2}, x_0 = 0$$ Then, prove that $$0le x_nle x_{n+1}le sqrt{x}$$

## If the sequence $x_n = dfrac {1} { sqrt {n}} left (1+ dfrac {1} { sqrt {2}} + dfrac {1} { sqrt {3}} + ldots + dfrac {1} { sqrt {n}} right)$ monotonous?

Keep that in mind $$x_1 = 1$$ and $$x_2 = dfrac {1} { sqrt {2}} left (1+ dfrac {1} { sqrt {2}} right)> dfrac {1} { sqrt {2}} left ( dfrac {1} { sqrt {2}} + dfrac {1} { sqrt {2}} right) = 1$$.

Consequently, $$x_2> x_1$$. In general we have too $$x_n = dfrac {1} { sqrt {n}} left (1+ dfrac {1} { sqrt {2}} + dfrac {1} { sqrt {3}} + ldots + dfrac {1} { sqrt {n}} right)> dfrac {1} { sqrt {n}} left ( dfrac {1} { sqrt {n}} + dfrac {1} { sqrt {n}} + dfrac {1} { sqrt {n}} + ldots + dfrac {1} { sqrt {n}} right) = 1$$.

Consequently, $$x_n geq 1$$ for all $$n in mathbb {N}$$. We also have

$$x_ {n + 1} = dfrac {1} { sqrt {n + 1}} left ( sqrt {n} x_n + dfrac {1} { sqrt {n + 1}} right) = dfrac { sqrt {n}} { sqrt {n + 1}} x_n + dfrac {1} {n + 1}$$.

Is it true that $$x_ {n + 1}> x_n$$?

## 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).

## 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.

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## 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.)

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## lebesgue integral – Prove that $Lambda (f) = int_ {0} ^ {1} frac {f (t)} { sqrt {| x-t |}}$ is a limited operator of $L ^ 2[0,1]$ to $L ^ 2[0,1]$

To prove $$Lambda (f) = int_ {0} ^ {1} frac {f (t)} { sqrt {| x-t |}}$$ is operator of limited $$L ^ 2 (0.1)$$ to $$L ^ 2 (0.1)$$

My idea was to use Jensens, but it doesn't seem to work.

$$int_ {0} ^ {1} {( int_ {0} ^ {1} frac {f (t)} { sqrt {| xt |}})} ^ 2 leq int_ {0} ^ {1} { int_ {0} ^ {1} frac {f (t) ^ 2} {| xt |}} = int_ {0} ^ {1} f (t) ^ 2 ln (1 – frac {1} {t})$$
But the logarithm is not in $$L ^ infty$$ and so I can't use a holder and finish the proof. I used Fubini and Jensen above.

How do I solve this problem? There is a hint that says to use $$sqrt {| x-t |} = sqrt (4) {| x-t |} sqrt (4) {| x-t |}$$ but I don't see how.

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## Asymptotics – Why doesn't $frac {n ^ 3} {2 ^ { Omega ( sqrt { log n})}}$ disprove the lower limit of $O (n ^ {3- delta})$?

I have a simple question:

All Pairs Shortest Path (APSP) is believed to have none $$O (n ^ {3- delta)}$$-Time algorithm for everyone $$delta> 0$$ by SETH.

Likewise

There is a result that says APSP can be solved in time $$frac {n ^ 3} {2 ^ { Omega ( sqrt { log n})}}$$ by Ryan Williams.

However, this improvement does not contradict the assumptions.

So I did the following: I'm comparing between $$lim_ {n -> infty} frac {( frac {n ^ 3} {2 ^ { sqrt { log n}})} {n ^ {3- delta}} = 0$$ In order to, $$frac {n ^ 3} {2 ^ { Omega ( sqrt { log n})}}$$ is better than the other, why doesn't that mean we refute the presumption.

If I have this function: $$frac {n ^ 3} {2 ^ { Omega ( sqrt { log n})}}$$I didn't know I should compare it to others because Big Omega is only part of it. How can you generally compare it to other features if you have them?

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## Real solutions for $sqrt {a ^ 2 + b ^ 2} – sqrt {1 + c ^ 2 + d ^ 2}> | a-c | + | b-d |$

Are there real numbers? $$a, b, c, d$$ so that
$$sqrt {a ^ 2 + b ^ 2} – sqrt {1 + c ^ 2 + d ^ 2}> | a-c | + | b-d |?$$

## Calculus – area of ​​a polar diagram in the form of $x$ and $y$ – $int sqrt {x ^ 2 + y ^ 2} d left ( tan ^ {- 1} frac {y} x {} right)$

I am new to polar coordinates and have not yet tried to find their area.

$$int sqrt {x ^ 2 + y ^ 2} d left ( tan ^ {- 1} frac {y} x {} right)$$

I just replaced it $$f ( theta)$$ With $$sqrt {x ^ 2 + y ^ 2}$$ and $$theta$$ With $$tan ^ {- 1} frac {y} x {}$$I thought there was a substitution that would allow me to express myself $$x$$ and $$y$$ as a relationship to one another – perhaps using logarithms. Is there a way to solve this integral?

## Power series – complex nested radicals ${ Re} Big ( sqrt {1+ frac {i} {2} sqrt {1+ frac {i} {2 ^ 2} sqrt {1+ frac {i } {2 ^ 3} sqrt {1+ frac {i} {2 ^ 4} sqrt { cdots}}}} Big) = 1$

One last question about nested radicals, but this time with a complex value:

$${ Re} Big ( sqrt {1+ frac {i} {2} sqrt {1+ frac {i} {2 ^ 2} sqrt {1+ frac {i} {2 ^ 3} sqrt {1+ frac {i} {2 ^ 4} sqrt { cdots}}}} Big) = 1$$

I tried to use power sets like Somos' answer, but I fail.

We also have a nice relationship
$$S = { Re} Big ( sqrt {1+ frac {i} {2} sqrt {1+ frac {i} {2 ^ 2} sqrt {1+ frac {i} { 2 ^ 3} sqrt {1+ frac {i} {2 ^ 4} sqrt { cdots}}}} Big) + { Im} Big ( sqrt {1+ frac {i} { 2} sqrt {1+ frac {i} {2 ^ 2} sqrt {1+ frac {i} {2 ^ 3} sqrt {1+ frac {i} {2 ^ 4} sqrt { cdots}}}} Big) = sqrt {1+ frac {1} {2} sqrt {1+ frac {1} {2 ^ 2} sqrt {1+ frac {1} {2 ^ 3} sqrt {1+ frac {1} {2 ^ 4} sqrt { cdots}}}} = 1.25$$
I also find it good to know what happened to Herschfeld's theorem in the case of a complex number. If someone has some paper on it, it would be cool.

My question: How do you solve it?