Achilles tendons still hurting... it's been 8 days now. I plan to swim today, that's it. Though I'm almost done putting together a strength-training regimen to start soon.
This morning on the way to camp, Jesse asked me what's bigger, a googol factorial or a googolplex. I told him I wasn't sure without paper, but he wanted me to guess.
Then I recalled that using a simple integral lowerbound, we know that (n/e)ne ≤ n!. (You get this from taking ln of n!, converting to a sum; then integrate ln x to get x ln x - x, and evaluate over [1,n] to get n ln n - n + 1. Taking e to this power gives the lowerbound (n/e)ne ≤ n!)
This makes it easy. Let g be a googol. Recall a googol is 10100 and a
googolplex is 10g. Then g! > (g/e)g e > (g/10)g = 1099g > 10g. So g! is definitely bigger than a googolplex. By a lot, in fact.
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