about-e - Something around the number e 1 n 1. Show that...

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Something around the number e 1. Show that the sequence  1 1 n n converges, and denote the limit by e . Proof: Since 1 1 n n k 0 n k n 1 n k 1 n 1 n n n 1 2! 1 n 2 .. n n 1  1 n ! 1 n n 1 1 1 2! 1 1 n ... 1 n ! 1 1 n  1 n 1 n 1 1 1 2 1 2 2 .. 1 2 n 1 ... 3, 1 and by (1), we know that the sequence is increasing. Hence, the sequence is convergent. We denote its limit e .Tha tis , lim n 1 1 n n e . Remark: 1. The sequence and e first appear in the mail that Euler wrote to Goldbach. It is a beautiful formula involving e i 1 0. 2. Use the exercise, we can show that k 0 1 k ! e as follows. Proof: Let x n 1 1 n n , and let k n ,wehave 1 1 1 2! 1 1 k .. 1 n ! 1 1 k  1 n 1 k x k which implies that ( let k ) y n : i 0 n 1 i ! e . 2 On the other hand, x n y n 3 So ,by(2)and(3) ,wefinallyhave k 0 1 k ! e . 4 3. e is an irrational number. Proof: Assume that e is a rational number, say e p / q , where g.c.d. p , q 1. Note that q 1. Consider
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about-e - Something around the number e 1 n 1. Show that...

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