Follow the above link or click the graphic below to visit the Homepage. 
Sums of Powers of the Natural Numbers Jim Cullen 
inf  
H =  1 i  = infinity  
i = 1 
n  
H_{ n } =  1 i  
i = 1 
n  
H_{ n } =  1 i  = Psi( n + 1 )  Psi( 1 )  
i = 1 
n  
H_{ n }  H_{m1} =  1 i  = Psi( n + 1 )  Psi( m )  
i = m 
H_{ n,2 }  =  1 ( 1 )^{2}  +  1 ( 2 )^{2}  +  1 ( 3 )^{2}  +  1 ( 4 )^{2}  +  1 ( 5 )^{2}  + ^{. . .} 
H_{ n,1 }  =  1 ( 1 )^{1}  +  1 ( 2 )^{1}  +  1 ( 3 )^{1}  +  1 ( 4 )^{1}  +  1 ( 5 )^{1}  + ^{. . .} 
inf  
S_{2} =  1 i^{ 2}  =  pi^{ 26}  
i = 1 



 
Power ( p )  Sum  Power ( p )  Sum  
1  Psi_{0}( n + 1 )  Psi_{0}( m ) 0 ! 
2  Psi_{1}( m )  Psi_{1}( n + 1 ) 1 !  
3  Psi_{2}( n + 1 )  Psi_{2}( m ) 2 ! 
4  Psi_{3}( m )  Psi_{3}( n + 1 ) 3 !  
5  Psi_{4}( n + 1 )  Psi_{4}( m ) 4 ! 
6  Psi_{5}( m )  Psi_{5}( n + 1 ) 5 !  
7  Psi_{6}( n + 1 )  Psi_{6}( m ) 6 ! 
8  Psi_{7}( m )  Psi_{7}( n + 1 ) 7 ! 
 
Power ( p )  Sum  Power ( p )  Sum  
1   Psi_{0}( m ) 0 ! 
2  Psi_{1}( m ) 1 !  
3   Psi_{2}( m ) 2 ! 
4  Psi_{3}( m ) 3 !  
5   Psi_{4}( m ) 4 ! 
6  Psi_{5}( m ) 5 !  
7   Psi_{6}( m ) 6 ! 
8  Psi_{7}( m ) 7 ! 
 
Power ( p )  Sum  Power ( p )  Sum  
1   Psi_{0}( 1 ) 0 ! 
2  Psi_{1}( 1 ) 1 !  
3   Psi_{2}( 1 ) 2 ! 
4  Psi_{3}( 1 ) 3 !  
5   Psi_{4}( 1 ) 4 ! 
6  Psi_{5}( 1 ) 5 !  
7   Psi_{6}( 1 ) 6 ! 
8  Psi_{7}( 1 ) 7 ! 
For even integer p > 0  
Z_{R}( p ) =  2^{ p1} abs( B_{p} ) pi^{ p} p ! 
inf  
Z_{R}( p ) =  1 i^{ p}  =  (1)^{ p} ^{.} Psi_{p1}( 1 ) ( p  1 ) !  
i = 1 
inf  
Z_{H}( p , a ) =  1 ( i + a )^{ p}  =  (1)^{ p} ^{.} Psi_{p1}( a ) ( p  1 ) !  
i = 0 
inf  
Z_{R}( s ) =  1 1  ( P_{ k } )^{  s }  
k = 1 
inf  
B_{ 2 n } =  2 ^{ . } ( 1 )^{ n1 }^{ . } ( 2 n ) ! ( 2 pi )^{ 2 n }  P^{ 2n }  
P = 1 
for even integer n > 38  
B_{ n } =  2 ^{ . } n ! ^{ . } (  1 )^{ ( n/21 )} ( 2 pi )^{ n } 
Psi_{ 0 }( x ) =  ln ( x^{ 2}  x ) 2  +  1 6 x^{ 2}  6 x +1 
Psi_{ 0 }( x + 1 ) = Psi_{ 0 }( x ) + x^{ 1 } 
Psi_{ 0 }(  x ) = Psi_{ 0 }( x ) + pi / tan [ pi ( x + 1 ) ] 
Psi_{ 0 }  =  ln ( x )    1 2 x    1 12 x^{ 2 }  +  1 120 x^{ 4 }    1 252 x^{ 6 }  +  1 240 x^{ 8 }    1 132 x^{ 10 }  + ^{. . .} 
inf  
Psi_{ 0 }( x ) = ln ( x )   1 2 x    B_{ 2 k } 2 k^{ . }x^{ 2 k }  
k = 1 
Besides the standard texts, perhaps the greatest online mathematical resource is Wolfram's Mathworld, from the makers of Mathematica software. Hardware support for much of the work on this page includes: Texas Instruments' Voyage200 PLT ( 12MHz M68000 CPU ) for summations, algorithmic work, numerical analysis, and some algebraic operations with the help of Bhuvanesh Bhatt's Mathtools calculator software; HewlettPackard's HP50G ( 75MHz ARM CPU ) for summations and some algebraic operations involving advanced functions; Microsoft QuickBasic Ver4.5 for generation of tables of highspeed numeric sums and crosschecking. The best stuff, as always, is worked out on pencil and paper ( CPU supplied by user ). 