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Exponentiation and the exponential increasing

Exponentiation and the exponential increasing

The exponentiation is the action of multiplying a number by itself a given number of times.

For instance, 3 to the power 2 is 3x3=9, 10 to the power 3 is 10x10x10=1,000.

A sequence starting wit u(0)=C and continuing with the iterative formula u(n+1)=a x u(n) has for value u(n) equal to C multiplied by a to the power n.

Such a sequence is called a geometrical sequence of initial point C and "reason" a>0.

There are 3 cases for the behavior of the sequence:

  • if a>1, the sequence goes to infinity in a very rapid manner, called "exponential increasing: the time by which it doubles is the same at each moment.
  • if a<1 (and a>0), the sequence goes to zeros in a quite slow manner, called "exponential decreasing": it takes the same time to divide the number by 2 and then to divide it by 2 again.
  • if a=1, the sequence is constant equal to u(0)=C.

Thus the behavior of a geometrical sequence is very unstable when the reason a varies in the neighborhood of 1: a little smaller than 1 gives going to 0 in exponential decreasing and a little bigger than 1 gives exponential increasing to infinity!

This is an example where the uncertainty of computation may lead to completely wrong results!

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