## imqwerty one year ago find general formula of --> 1^x +2^x +3^x.........n^x

1. imqwerty

@dan815

2. anonymous

$\sum_{a=1}^{n}a^{x}$

3. imqwerty

@MayankD i want to find formula like we have- $\sum_{n=1}^{n}n^2 =\frac{ n(n+1)(2n+1) }{ 6}$

4. dan815

hey

5. imqwerty

hey :)

6. dan815

oaky so if u noticed this u can rewrite this in some slightly different notation like theyre all just sum of sums

7. dan815

|dw:1443861264405:dw|

8. dan815

|dw:1443861345422:dw|

9. dan815

you see how these sums of sums is working

10. dan815

now can we get a general form for sum of sums formula

11. imqwerty

why did u say its =0N^2 ?

12. dan815

it means the highest degree for our closed formula is of degree 2

13. imqwerty

ok :)

14. dan815

jsut simplifying the math for now, we can worry about the additional order terms later

15. anonymous

Big-O is a function which defines an upper bound for another function

16. anonymous

Asymptotically that is

17. dan815

now lets thinking if there is some way of computing sum of sum of sum of sums...

18. dan815

|dw:1443861532777:dw|

19. imqwerty

cx

20. anonymous

I do not think you will get a simplistic answer like those you showed me for this variable exponential summation. The answer has to do with generalised harmonic numbers(courtesy wolfram alpha)

21. imqwerty

ye i saw that too :)

22. dan815

yeah xD i think this is why i stopped after i got the recursive equation to keep generating 1 degree higher

23. dan815

but hey i think there is still some ways

24. dan815

to get approximate solutions

25. imqwerty
26. dan815

this stuff reminds me of this sigma function thing

27. dan815

which is used in arithmetic function stuff, and u do something called dirichlet convolutions

28. dan815

and i believe they are also formed out of the harmonic series

29. dan815

take a look at this if you are interested https://www.youtube.com/watch?v=X0XJ3TuMiFc

30. Zarkon

|dw:1443896724330:dw| this is not true

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