If the series from n=1 to infinity of n^P converges, then which of the following is false?

a. P < 1
b. the series from n=1 to inf of n^-P converges
c. the integral from n=1 to inf of x^P dx is finite
d. the integral from n=1 to inf of x^(P-1) dx is finite
e. the integral from n=1 to inf of x^(P-2) dx is finite

I think the answer is b, because P must be less than -1 to converge, and -P would just be a positive P that would lead the series to diverge.

To determine which statement is false, we need to analyze the convergence requirements of the series from n=1 to infinity of n^P.

The given series converges if P < -1. Therefore, option (b) states that the series from n=1 to infinity of n^-P converges. This is true since when we take the reciprocal, -P becomes positive and satisfies the convergence condition. So, option (b) is correct.

Now, let's consider the remaining options:

a. P < 1:
This statement is true. For the series to converge, the exponent P must be less than -1, not just less than 1. So, option (a) is correct.

c. The integral from n=1 to infinity of x^P dx is finite:
This statement is true. If the series converges, then the area under the curve defined by the function x^P also converges. So, option (c) is correct.

d. The integral from n=1 to infinity of x^(P-1) dx is finite:
This statement is false. The integral from n=1 to infinity of x^(P-1) dx also converges, given that the series from n=1 to infinity of n^P converges. So, option (d) is incorrect.

e. The integral from n=1 to infinity of x^(P-2) dx is finite:
This statement is true. Since the integral from n=1 to infinity of x^(P-1) dx converges, the integral from n=1 to infinity of x^(P-2) dx will also converge. So, option (e) is correct.

Therefore, the false statement is option (d), as explained above.