A. if a new series is found by differentiating a known series with included endpoints, those endpoints might not be included in the new series.
b. if a new series is found by differentiating a known series with included endpoints, those endpoints will never be included in the new series.
which statements are false?

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.

The answer is biodiversity resulting from new ancestors.