**Ian finds patterns in the ratios
and differences **

** of powers of powers, and gets
an infinite sequence that approaches e.**

A little background: From the 2nd grade when Ian first came to The Math Program, he was always looking for patterns, using differences and ratios. When Ian was here in about his 10th year in The Math Program, he worked on powers of powers. Don gave him a copy of the article "Polypowers" from Martin Gardner's book "Knotted Doughnuts and other Mathematical Entertainments" (published by W.H. Freeman and Company, NY, 1986). Subsequently, Ian was playing around with his calculator in his Physics class. Here is the result of this "playing around".

In line 1 below, when x = 2, he found (2+1)^{(2+1) }=
3^{3 }= 27 and for line 2 when x = 3,
he got (3+1)^{(3+1) }= 4^{4 }= 256 and so on.

Ian saw that
the differences where getting very big, like the powers themselves. So he
decided to look at the **ratios** of the powers. He
noticed that the ratios do not get
very big and the **differences** of these **ratios**
might lead to something.

He noticed that these differences on the right formed an **infinite
sequence** approaching **e**.
So his function is and the
limit as x goes to infinity or = **e
(WOW!)**

Ian graphed his function in *Mathematica*,

an infinite sequence which approaches 2.71828... which = **e**

**Great job Ian!!**

Don had never seen this before, nor since.

Notice: **e **is also
defined as the limit_{n->inf}(1+1/n)^{n }
[read this as "the limit of (1+1/n)^{n }as n goes to
infinity"] and can be
written as an infinite series

See Geoffrey's work with powers of powers.

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