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⁾ = 3³ = 27 and for line 2 when   x = 3, he got (3+1)⁽³⁺¹⁾ = 4⁴ = 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 were getting close to e. So his function is and the limit as x goes to infinity or = e   (WOW!)

Don had never seen this before, nor since.

See Geoffrey's work with powers of powers.

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