For which #x in RR# does the series P(x) converge and for which does it diverge? #P(x) = sum 1/n * x^(n+2)# And how do i show that the relation #P'(x) - 2*((P(x))/x) = x^2/(1-x)# is valid for the inner side of of the convergence intervall?
1 Answer
Convergence interval is
Explanation:
The
Thus we have
so that
and according to the ratio test, the series will converge for
(it is well known that for
Now, within the interval of convergence, a power series can be differentiated term by term, so that
And thus
The final result is an infinite geometric series with first term