perplexus.info :: Calculus : Multiply together, get Euler?

Multiply together, get Euler? (Posted on 2007-04-24)

S_p = p (1 + S_p-1) for p>=2 and S₁=1.

Determine, whether or not:

Limit (1+ 1/S₁)(1+ 1/S₂)(1+ 1/S₃).....(1+ 1/S_p)= e
p-> infinity

where e is the Euler's Number.

Submitted by K Sengupta

Rating: 3.5000 (2 votes)

Solution: (Hide)

We observe that:
S_p
= p + p*S_p-1
= p + p(p - 1) + p(p - 1)S_p-2
= p + p(p - 1) + p(p - 1)(p - 2) + p(p - 1)(p - 2)S_{p-3_{= ............

= .........

= p + p(p - 1) + p(p - 1)(p - 2)+ .........+ p(p - 1)(p - 2)......(3)(2)
+ p(p - 1)(p - 2)......(3)(2)S₁

Since S₁ = 1, it follows that:

S_p = p!/(p - 1)! + p!/(p-2)! + ... + p!/1! + p!/0!

= p!*Sum ( k = 0 to p-1) (1/k!)

We now observe that, p!/p! = 1 and, accordingly:

S_{p + 1} = p! Sum(k = 0 to p) (1/k!).

Consequently:

1 + 1/S_p

= (S_p + 1)/S_p

= [Sum(k = 0 to p) 1/k!]/[Sum(k = 0 to p - 1) 1/k!].

Thus,

Limit (1+ 1/S₁)(1+ 1/S₂)(1+ 1/S₃).....(1+ 1/S_p)

p-> infinity

= [Sum(k = 0 to infinity) 1/k!]/[1/0!]

= e}}

Comments: ( You must be logged in to post comments.)

	Subject	Author	Date
	computer test	Charlie	2007-04-24 09:52:40

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