Thread: Help with evaluating indefinite integral as power series?

How would I solve this problem using power series?

Integral of [ln(1-t)]/t dt? I know that if I find the power series for [ln(1-t)]/t, then I could integrate each term separately and then that would be the solution, but how do I find the power series for such an expression? I know that the derivative of ln(x) is 1/x and the integral of 1/x is ln(x), and it seems like that would somehow apply in this problem, but I'm not sure how. Help please?

Thanks.

Originally Posted by virtuoso735

How would I solve this problem using power series?

Integral of [ln(1-t)]/t dt? I know that if I find the power series for [ln(1-t)]/t, then I could integrate each term separately and then that would be the solution, but how do I find the power series for such an expression? I know that the derivative of ln(x) is 1/x and the integral of 1/x is ln(x), and it seems like that would somehow apply in this problem, but I'm not sure how. Help please?

Thanks.

Let

so we can take the derivative to get



now we can integrate this to find a series for f(t)



so we can use this to find the series we need



You should be able to finish from here...