perplexus.info :: Calculus : Factorial Ratio Sum Puzzle

Factorial Ratio Sum Puzzle (Posted on 2015-08-04)

Define:
A(k) = k! + (k+1)! + (k+2)!, and:

B(n) = Σ_{k=1 to n} (k+2)/A(k)

Evaluate:

Limit B(n)
n → ∞

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)

more demonstration than proof Comment 3 of 3 |

A(k)=k!*(1 + k+1 + (k+1)(k+2)) = k!*(k+2)^2

Then the n-th term of B is 1/(n!*(n+2))

The nth term of B is <= 1/2^(n+1)
** (equal when n=2)
so limit B(n) <= 1/4 + 1/8 + 1/16 + . . . = 1/2 = .5

The nth term of B is also >= 1/3^n
** (equal when n=1)
so limit B(n) >= 1/3 + 1/9 + 1/27 + . . . = 1/2 = .5

and thus limit B(n)=.5

Posted by xdog on 2015-08-04 10:30:53

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