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Math Help - nth term for series

  1. #1
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    Post nth term for series

    Hey all!

    I needed some help in how to get the nth term (Un) for some series, that are:

    1/1 + 3/2 + 5/2^2 + 7/2^3 + ...

    1/2 + 2/3 + 3/4 + 4/5 + ...

    1/5 + 2/6 + 2^2/7 + 2^3/8 + 2^4/9 + ...

    I know the nth term (Un) for each but I wanted to know how you get them.

    Thank you
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  2. #2
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    Quote Originally Posted by dadon View Post
    Hey all!

    I needed some help in how to get the nth term (Un) for some series, that are:

    1/1 + 3/2 + 5/2^2 + 7/2^3 + ...

    1/2 + 2/3 + 3/4 + 4/5 + ...

    1/5 + 2/6 + 2^2/7 + 2^3/8 + 2^4/9 + ...

    I know the nth term (Un) for each but I wanted to know how you get them.

    Thank you
    Basicaly by inspection, and making some assumptions.

    Look at the first:

    1/1 + 3/2 + 5/2^2 + 7/2^3 + ...

    The numerators are consecutive odd numbers, which we may write as 2k-1
    for k=1, 2, .. The denominators are consecutive powers of 2 which we may
    write 2^(k-1) k=1, 2, .. (the -1 in the exponent is required here as I am
    taking the first term to correspond to k=1 for both the numerator and
    denominator), so the k-th term is (2k-1)/2^{k-1}.

    Now look at the second:

    1/2 + 2/3 + 3/4 + 4/5 + ...

    The numerators are consecutive integers which we may write k, k=1, 2, ..
    The denominators are also consecutive integers, but starting from 2, so
    may be written (k+1), k=1, 2, .. Hence the general term is k/(k+1).

    RonL
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  3. #3
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    Hello, dadon!

    These have pretty obvious patterns . . .


    1/1 + 3/2 + 5/2 + 7/2 + ...

    The numerators are: 1, 3, 5, 7, ...
    . . The general numerator is: 2n - 1

    The denominators are: 2^
    0, 2^1, 2^2, 2^3, ...
    . . The general denominator is: 2^
    {n-1}

    The general term is: .a
    n .= .(2n - 1) / 2^{n-1}


    1/2 + 2/3 + 3/4 + 4/5 + ...

    The numerators are: 1, 2, 3, 4, ...
    . . The general numerator is: n

    The denominators are: 2, 3, 4, 5, ...
    . . The general denominator is: n + 1

    The general term is: .a
    n .= .n / (n + 1)


    1/5 + 2/6 + 2/7 + 2/8 + ...

    The numerators are: 2^
    0, 2^1, 2^2, 2^3, ...
    . . The general numerator is: 2^{n-1}

    The denominators are 5, 6, 7, 8, ...
    . . The general denominator is: n + 4

    The general term is: .a
    n .= .2^{n-1} / (n + 4)

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  4. #4
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    Post thanks

    Thank you!

    Yeah does seem quite straight forward now. I was getting confused on the 2^(n-1) and was forgetting the 2^0, 2^1 terms.

    Thanks again

    Kind regards,

    Dadon
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