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Power Summations #2
(Math | Calculus | Series | Power2)

Summation Expansion Equivalent Value Comments
  inf
sum 1/n
n=1
= 1 + 1/2 + 1/3 + 1/4 + ... diverges to inf see the gamma constant
  inf
sum 1/n 2
n=1
= 1 + 1/4 + 1/9 + 1/16 + ... = (1/6) PI 2 = 1.64493406684822... see Expansions of PI
  inf
sum 1/n 3
n=1
= 1 + 1/8 + 1/27 + 1/81 + ... = 1.20205690315031... see the Unproved Theorems
  inf
sum 1/n 4
n=1
= 1 + 1/16 + 1/81 + 1/256 + ... = (1/90) PI 4 = 1.08232323371113... see Expansions of PI
  inf
sum 1/n 5
n=1
= 1 + 1/32 + 1/243 + 1/1024 + ... = 1.03692775514333... see the Unproved Theorems
  inf
sum 1/n 6
n=1
= 1 + 1/64 + 1/729 + 1/4096 + ... = (1/945) PI 6 = 1.017343061984449... see Expansions of PI
  inf
sum 1/n 7
n=1
= 1 + 1/128 + 1/2187 + 1/16384 + ... = 1.00834927738192... see the Unproved Theorems
  inf
sum 1/n 8
n=1
= 1 + 1/256 + 1/6561 + 1/65536 + ... = (1/9450) PI 8 = 1.00407735619794... see Expansions of PI
  inf
sum 1/n 9
n=1
= 1 + 1/512 + 1/19683 + 1/262144 + ... = 1.00200839282608... see the Unproved Theorems
  inf
sum 1/n 10
n=1
= 1 + 1/1024 + 1/59049 + 1/1048576 + ... = (1/93555) PI 10 = 1.00099457512781... see Expansions of PI
  inf
sum 1/(2n) n
n=1
= 1 + 1/(2n) 2 + 1/(2n) 3 + 1/(2n) 4 + ... = (-1)n-1 ( 2 2n B(2n) PI 2n ) / ( 2(2n)! ) see Expansions of PI

  
 
  

 
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