import 4.code.options;
import 4.code.about;

class Header{

public void title(){

String fullTitle = "// - ";
}

public void menu();

public void board();

public void goToBottom();

public void refresh(a);

}
class Thread extends Board{
public void EmptyTitle(OP Anonymous){

String fullTitle = "EmptyTitle";
int postNumber = "575412";
String image = "Capture.png";
String date = "10/10/18(Wed)14:24:50";
String comment = "How is this even considered a question? this is 2nd year cs proofs btw. would like hints more than just the full answer but idk where to even start. do i find a c1, c2 that bound the fxn? how do i do that and then what. im totally lost";

}
public void comments(){
if(Anonymous && title=="" && postNumber==575452 && dateTime=="10/10/18(Wed)17:15:01")

"You can see immediately that the sum is less than n^{k+1} because there are n terms, each bounded by n^k.
On the other hand, the sum is greater than n * (n/2)^k. = 2^{-k} n^{n+1}. So you just have to prove that."
;


if(Anonymous && title=="" && postNumber==575454 && dateTime=="10/10/18(Wed)17:25:02")

">>575452
More details (don't read if you don't want spoilers):

Prove that
(a-i)^k + (a+i)^k >= 2a^k
by expanding the terms on the left using the Binomial theorem. Use this to prove that the original sum
1^k + 2^k + ... + (n/2)^k + ... + ... + (n-1)^k + n^k
is greater than
(n/2)^k + (n/2)^k + ... + (n/2)^k + ... + (n/2)^k + (n/2)^k
= n * (n/2)^k.

I assume you understand what Θ means."
;


if(Anonymous && title=="" && postNumber==575593 && dateTime=="10/11/18(Thu)02:11:37")

">>575454
>I assume you understand what Θ means.


uh... the... Eye of Sauron?"
;


if(Degenerate && title=="" && postNumber==575608 && dateTime=="10/11/18(Thu)03:10:08")

"What language is this?";


if(Anonymous && title=="" && postNumber==575610 && dateTime=="10/11/18(Thu)03:15:56")

">>575608
math"
;


if(Anonymous && title=="" && postNumber==576058 && dateTime=="10/12/18(Fri)03:27:06")

">>575452
> 2^{-k} n^{n+1}

Dat is een oef"
;


if(Anonymous && title=="" && postNumber==576059 && dateTime=="10/12/18(Fri)03:30:34")

">>575593
>>I don't understand what Θ means
If you didn't know that it was an uppercase theta, that means you're incapable of using google and therefore not fit to be a CS major. In which case, kill yourself."
;


if(Anonymous && title=="" && postNumber==576061 && dateTime=="10/12/18(Fri)03:33:08")

">>576058
>2^{-k} n^{n+1}
You're right, it should say 2^{-k} n^{k+1}.

>>576059
pls no bully"
;


if(Anonymous && title=="" && postNumber==576087 && dateTime=="10/12/18(Fri)04:20:55")

">>575412
for the lower bound it's easier to throw away the first half of the terms < (n/2)^k and bound the remaining by below by (n/2)^k, so you obtain:
sum >= (n/2)(n/2)^k = 2^(-k-1) n^(k+1)"
;


}
}