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When the base
is equal to the root index
we get
![{\displaystyle {\sqrt[{n}]{n}}=n^{\frac {1}{n}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/17cba321c001db01540bbf06ab8cfe1a610e74ed)
which may tentatively be notated n//n, being the inverse operation of n**n (i.e. n^n.)
Formulae
Recurrence relation
Generating function
Differences
Partial sums
Partial alternating sums
Alternating series
The alternating series
![{\displaystyle \sum _{n=1}^{\infty }(-1)^{n}~n^{1/n}=\sum _{n=1}^{\infty }(-1)^{n}~{\sqrt[{n}]{n}}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/f4150123bf21f688bc83b04103b6e2df485cf8ef)
gives an oscillating divergent series whose upper limit point is the MRB constant, the lower limit point being the MRB constant - 1.
Partial sums of reciprocals
Sum of reciprocals
See also
Hierarchical list of operations pertaining to numbers [1] [2]
0th iteration
1st iteration
- Addition:
S(S(⋯ "a times" ⋯ (S(n)))) |
, the sum , where is the augend and is the addend. (When addition is commutative both are simply called terms.)
- Subtraction:
P(P(⋯ "s times" ⋯ (P(n)))) |
, the difference , where is the minuend and is the subtrahend.
2nd iteration
- Multiplication:
n + (n + (⋯ "k times" ⋯ (n + (n)))) |
, the product , where is the multiplicand and is the multiplier.[3] (When multiplication is commutative both are simply called factors.)
- Division: the ratio , where is the dividend and is the divisor.
3rd iteration
- Exponentiation ( as "degree", as "base", as "variable").
- Powers:
n ⋅ (n ⋅ (⋯ "d times" ⋯ (n ⋅ (n)))) |
, written .
- Exponentials:
b ⋅ (b ⋅ (⋯ "n times" ⋯ (b ⋅ (b)))) |
, written .
- Exponentiation inverses ( as "degree", as "base", as "variable").
4th iteration
- Tetration ( as "degree", as "base", as "variable").
- Tetration inverses ( as "degree", as "base", as "variable").
5th iteration
- Pentation ( as "degree", as "base", as "variable").
- Pentation inverses
6th iteration
- Hexation ( as "degree", as "base", as "variable").
- Hexation inverses
7th iteration
- Heptation ( as "degree", as "base", as "variable").
- Heptation inverses
8th iteration
- Octation ( as "degree", as "base", as "variable").
- Octa-powers:
n ^^^^^ (n ^^^^^ (⋯ "d times" ⋯ (n ^^^^^ (n)))) |
, written .
- Octa-exponentials:
b ^^^^^ (b ^^^^^ (⋯ "n times" ⋯ (b ^^^^^ (b)))) |
, written .
- Octation inverses
Notes
Notes