Gelfond’s Constant

Simon spent two days testing out his new Texas TI-84 Plus CE-T calculator. I saw him play with Gelfond’s constant eπon the calculator:

He looked up on Wikipedia that the decimal expansion of Gelfond’s constant begins as follows:

e^{\pi }\approx 23.14069263277926900572908636794854738\dots \,.

And that if one defines {\displaystyle k_{0}={\tfrac {1}{\sqrt {2}}}} and

{\displaystyle k_{n+1}={\frac {1-{\sqrt {1-k_{n}^{2}}}}{1+{\sqrt {1-k_{n}^{2}}}}}}

for n>0, then the sequence

{\displaystyle (4/k_{n+1})^{2^{-n}}}

converges rapidly to e^{\pi }.

I then saw Simon jot the formulas on the whiteboard, also using limits (something he came up with himself and not looked up). Glowing, he announced he was going to try to calculate Gelfond’s constant by hand:


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