Sequences - monotonic or not

  1. Now I know how this works- but I came across this example and even though I know the answer- the simplification given in the explaination doesn't make sense to me.

    the squence is an= {5n/n!}
    now applying an+1 and dividing an+1/an
    the book indicates = 5/n+1

    this is what I don't get how
    (5n+1 /(n+1)!)/(5 n/n!) can simplify to that ?

    can someone explain please- what am I missing here.
     
    Last edited: Apr 2, 2012
  2. jcsd
  3. We have...
    [tex]\frac{5^{n+1}}{(n+1)!}\frac{n!}{5^n} = \frac{5\cdot5^{n}}{(n+1)n!}\frac{n!}{5^n}[/tex]
    ...which very easily simplifies to the expression you provided by cancelling out like terms.
     
  4. right - this is what is not clear to me- I am very new to pure maths
    how (n+1)! can be written as - (n+1)n!- may be I am having a dumb moment
     
    Last edited: Apr 2, 2012
  5. micromass

    micromass 19,347
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    What is the definition of n! for you?
     
  6. well n! means = any number say 5 then multiplied by 5x4x3x2x1 ( natural numbers in hughest to lowest order)
     
  7. so basically product of positive integres less than or equal to n
     
  8. micromass

    micromass 19,347
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    So, you have

    [tex](n+1)!=(n+1)*n*(n-1)*(n-2)*...*3*2*1[/tex]

    Right?

    But then we have

    [tex](n+1)!=(n+1)*[n*(n-1)*(n-2)*...*3*2*1][/tex]

    And the thing in brackets look familiar, no?? Indeed, the bracketed thing is n!
    So

    [tex](n+1)!=(n+1)*n![/tex]
     
  9. thank you this makes sense- sometimes I just get frustrated with not enough explaination at beiggners level
     
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