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Math Help - derive a formula for infinite series..

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    derive a formula for infinite series..

    Derive a general formula for the coefficients bn, as defined by (1/(1-x)){summation to infinity with i=0}(ai)x^i = {summation to infinity with i=0}(bi)x^i
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    Quote Originally Posted by carpark View Post
    Derive a general formula for the coefficients bn, as defined by (1/(1-x)){summation to infinity with i=0}(ai)x^i = {summation to infinity with i=0}(bi)x^i
    For \, -1 < x < 1\, , \, \frac{1}{1-x} = 1 + x + x^2 + x^3 + .......\, using the formula in reverse for an infinite geometric series.

    For \, x > 1\, or \, x < -1\,, \, \frac{1}{1-x} = -\frac{1}{x} \left( \frac{1}{1 - \frac{1}{x}}\right) = -\frac{1}{x} \left( 1 + \frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3} + ....\right) \, again using the formula in reverse for an infinite geometric series.
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    What if for each i>=0, ai=i, then what is bn?
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