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 $?

ordinary differential equations – what is the singular point of the ode $ frac {dy} {dx} = sqrt {y} + 1 $

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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.)

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.

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?

Thanks in advance!

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?

Thank you in advance .