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e and pi (Posted on 2003-09-26) Difficulty: 4 of 5
Without finding the numerical values, show which is greater, e^π or π^e.

See The Solution Submitted by DJ    
Rating: 4.1429 (14 votes)

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Solution By power series | Comment 4 of 7 |
The power series for e^x is 1 + x + x²/2! + x³/3! + ...
So we have e^x > 1 + x, for x > 0. (1)

Now set x = π/e - 1.
Then, by (1), e^(π/e - 1) > π/e.
Multiply by e: e^(π/e) > π.
Raise to the power e: e^π > π^e.
  Posted by Nick Hobson on 2003-10-09 17:08:14
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