How does the convergence or divergence of a series affect the possibility of bounded summation?

2n! Consider the infinite series defined by - 22 2. (a) What is the value of r from the ratio test? (b) What does this r value tell you about the series? Answer:

5. Consider the series n! n=0 for real numbers x. (a) Use the ratio test to determine the values x for which the series is absolutely convergent. (b) Does this mean that the series is convergent for these values of x? Explain. (c) Explain how one can use part (b) above to determine the limit of the sequence {an} where 22n an n!

Consider the series (a) Does the sequence (a,) =-converge or diverge? (b) Does this series converge or diverge? (c) How many terms are required for the sum to exceed 50?