Tetration

In mathematics, especially number theory, it is called a power tower, if the exponent of a power even more times is potentiated and thus add up the exponents to a tower. The notation is usually used for numbers in which the exponent would be too large in the normal mode, for example:

The larger the number is, the more distinct the shortening advantage of this notation.

It should be noted that power towers are processed from top to bottom, as

The exponent of this number would have expressed in normal notation, 19,728 points. The number would thus hardly usable or understandable. Using this notation, very large numbers can be represented, which are quickly beyond any direct imaginability and no longer or only with difficulty can represent in absolute length and as simple power.

Nevertheless, there are numbers that are so large that even this spelling is no longer sufficient to represent them. Thus, if a power tower has too many steps, as that they could still represent one uses alternative spellings such as the Hyper- operator.

Representation of sequences and infinite power towers

A finite power tower of the form ( see also arrow notation)

With and agrees with the result element of the sequence match:

The sequence itself is referred to as Partialturmfolge and identified with the infinite power tower (similar to the concept of infinite series ).

Is convergent with the limit value is then called the (infinite ) power tower with convergent

Leonhard Euler already has recognized that the power of the tower

If and only converges if

Is a then: