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    Right, this example is from a series review we're working on in Diff Eq and it's something my previous calc professor neglected to require us to know.

    " Rewrite the given expression as a sum whose generic term involves xn"

    (1-x^2)\displaystyle\sum_{n=2}^\inf  ty n(n-1)a_nx^{n-2}

    What I tried:

    (1-x^2)[2(1)a_2+3(2)a_3x+4(3)a_4x^2+5(4)  a_5x^3+6(5)a_6x^4+...]
    [2a_2(1-x^2)+6a_3(x-x^3)+12a_4(x^2-x^4)+20a_5(x^3-x^5)+30a_6(x^4-x^6)+...]

    But from here I can't see how that would simplify to something distinguishable with xn in it. Have I attempted it totally wrong or am I just not seeing a trick? Thanks.

    edit: Should I just focus on the summation and ignore the (1-x2)?

    *tries that*
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    (1-x^2) \sum n(n-1)a_nx^n = \sum n(n-1)a_nx^n - x^2 \sum n(n-1)a_nx^n = \sum n(n-1)a_nx^n - \sum n(n-1)a_n x^{n+2}
    Now sum m = n-2 in the 2nd sum.
 
 
 
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Updated: July 15, 2009
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