Thread: 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

Originally Posted by carpark

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 $\displaystyle \, -1 < x < 1\, $, $\displaystyle \, \frac{1}{1-x} = 1 + x + x^2 + x^3 + .......\, $ using the formula in reverse for an infinite geometric series.

For $\displaystyle \, x > 1\, $ or $\displaystyle \, x < -1\,$, $\displaystyle \, \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.

What if for each i>=0, ai=i, then what is bn?