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Thread: Why is this like this? Ratio Test on Series

  1. #1
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    Why is this like this? Ratio Test on Series

    Why is this like this? Ratio Test on Series-capture.png

    When i plug in the (n+1), the bottom isnt 5*8*11...(3(n+1)+2) but it is actually 5*8*11...(3n+2)*(3(n+1)+2)

    how come?
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    Re: Why is this like this? Ratio Test on Series

    Quote Originally Posted by lc99 View Post
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    When i plug in the (n+1), the bottom isnt 5*8*11...(3(n+1)+2) but it is actually 5*8*11...(3n+2)*(3(n+1)+2)

    how come?
    Maybe the (3n+2) is the factor just before (3(n+1)+2)????

    In the original expression the next to the last term was 3(n-1) +2. Just because it is not listed doesn't mean it does not exist. Now when YOU replace n with n+1 what do you get??
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    Re: Why is this like this? Ratio Test on Series

    Hey, when i replaced the n in (3n+2) with n+1, and also the n's in the numerator, i get that the limit is infinite which does not conclude anything about the original series. However, my professor has her answer key and it shows that she added a (3n+2) in the denominator and replaced THAT n with n+1
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    Re: Why is this like this? Ratio Test on Series

    Quote Originally Posted by lc99 View Post
    However, my professor has her answer key and it shows that she added a (3n+2) in the denominator and replaced THAT n with n+1
    Yes! I agree that you should replace ALL the n's with n+1. This includes the last term and the term before the last one.....
    Thanks from lc99
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    Re: Why is this like this? Ratio Test on Series

    Quote Originally Posted by lc99 View Post
    Click image for larger version. 

Name:	Capture.PNG 
Views:	6 
Size:	8.4 KB 
ID:	38664

    When i plug in the (n+1), the bottom isnt 5*8*11...(3(n+1)+2) but it is actually 5*8*11...(3n+2)*(3(n+1)+2)

    how come?
    5*8*11...(3(n-1)+2)*(3n+2). Let's just look at the last 2 terms. Now we replace n with n+1 and get (3((n+1)-1)+2)*(3(n+1)+2) = (3(n)+2))*(3(n+1)+2) = (3n+2))*(3(n+1)+2)
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    Re: Why is this like this? Ratio Test on Series

    Quote Originally Posted by lc99 View Post
    Click image for larger version. 

Name:	Capture.PNG 
Views:	6 
Size:	8.4 KB 
ID:	38664
    When i plug in the (n+1), the bottom isnt 5*8*11...(3(n+1)+2) but it is actually 5*8*11...(3n+2)*(3(n+1)+2)
    how come?
    Because this is an alternating series none of the usual test work.
    The non alternating part is $a_n=\dfrac{2^n\cdot n!}{5\cdot 8\cdots (3n+2)}$.
    If the sequence $a_n$ is decreasing to zero then the series converges.
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  7. #7
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    Re: Why is this like this? Ratio Test on Series

    ahhh. i get it now. thank you! i just never seen the type of series with a sequence like that. and combine thay with convergence tests
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