18. What are all the values of x for which the infinite series (x-5) + 2(x-5)2 + 3(x-5)3 + 4(x-5)4 + … converges?

 

A.

B.

C.

D.

E.

 

Solution: The answer is A

 

Rewrite the given series as:

By the ratio test,this series will converge if |x-5| < 1. Thatis, for
4 < x < 6. Furthermore, the series does not converge at the end points of the interval, 4 or 6, so (4, 6) is the interval of convergence.