Power Sum and Sum of Partial Power Sums

Power Sum and Sum of Partial Power Sums
(May 1, 2008)
 
Power Sum
 
     
      where p, n are positive integers.
 
 
A family of Power Sum formulas are formed for p = 1, 2, ..., 14 as shown in the following:
 
1.     
    
 
 
 
2.     
     
 
 
 
3.     
      
 
 
 
4.     
      
 
 
 
5.     
     
 
 
6.     
       
 
 
 
7.     
     
 
 
 
8.     
     
 
 
9.     
      
 
 
10. 
       
 
 
 
11. 
      
 
 
12. 
 
      
 
 
                       
 
13. 
      
 
 
 
 
14. 
       
 
 
 
 
Sum of Partial Power Sum (Update of Repeated Power Series, June 27, 2005)
 
Define
,
 
where m, n, k, and i are positive integers.
 
 
 
Computing the Sum of Partial Power Sum for m = 1, 2, …, 12 gives
 
1.     
 
 
 
2.     
 
 
 
3.     
 
 
 
4.     
 
 
5.     
 
 
 
6.     
 
 
 
7.     
 
 
 
8.     
 
 
 
9.     
 
 
10.     
 
 
 
 
11.     
 
 
 
 
12.     
 
 
 
 
(May 1, 2008)
 

Similar seriesFinite Alternative Odd Power Series.

In-Text or Website Citation
Tue N. Vu, Power Sum and Sum of Partial Power Sums, from Series Math Study Resource.
Hyperlink: http://seriesmathstudy.com/sms/PowerSumandSumofPowerSum.