Consider the power series
By the ratio test, this series converges for
though we know by the alternating series test that the series converges for
.
So this series converges for
.
Differentiating the series yields
which is the geometric series. We know this series converges for
, and this time the endpoints are not included.
This example shows that (A) is certainly possible; that is, is valid in the first series, but not in the differentiated one.
- - -
Now consider the series
which we know to converge to
.
Differentiating, we get
as expected. But both series converge everywhere, so this serves as a counter-example to the claim of B. So B is false.
By the ratio test, this series converges for
though we know by the alternating series test that the series converges for
So this series converges for
Differentiating the series yields
which is the geometric series. We know this series converges for
This example shows that (A) is certainly possible; that is,
- - -
Now consider the series
which we know to converge to
Differentiating, we get
as expected. But both series converge everywhere, so this serves as a counter-example to the claim of B. So B is false.