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Math Help - Infinite Serie

  1. #1
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    Infinite Serie

    If the nth partial sumof an infinite series a_n is s_n = 3 - n2^-n
    find a_n and the infinite series a_n
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  2. #2
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    Hello, amiv4!


    If the n^{th} partial sum of an infinite series is: . S_n \:= \:3 - n^2-n

    find a_n . . . and the infinite series a_n ?? .
    The series diverges
    Crank out the first few partial sums . . .

    . . \begin{array}{ccc} S_1 &=& 1 \\ S_2 &=& \text{-}3 \\ S_3 &=& \text{-}9 \\ S_4 &=& \text{-}17 \\ S_5 &-& \text{-}27 \\ \vdots &&\vdots \end{array}


    We can see that the summation is:

    . . S_n \;=\;1 - 4 - 6 - 8 - 10 - \hdots - 2n

    . S_n \;=\;\overbrace{3 - 2} - 4 - 6 - 8 - \hdots - 2n

    . S_n \;=\;3 -2(1 + 2 + 3 + 4 + \hdots + n)

    . S_n \;=\;3 - 2\,\frac{n(n+1)}{2} \quad\hdots\quad\text{which equals: }3 - n^2-n\quad\hdots YAY!


    I can't find a neat representation for a_n

    . . a_n \;=\;\bigg\{\begin{array}{ccc}1 & & n = 1 \\ \text{-}2n & & n > 1 \end{array}

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