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