Which of the following series could be tested for convergence/divergence with the integral test?
the summation from n=1 to infinity of 1/n!
the summation from n=1 to infinity of 1/n
the summation from n=2 to infinity of ln(n)/n^2
a) I only
b) II only
c) II and III -------> My answer. Can you check for me, please?
d) I and II
To determine which series could be tested for convergence or divergence using the integral test, we need to consider the conditions required for the integral test to be applicable.
The integral test can be used when both of the following conditions are met:
1. The terms of the series are positive.
2. The terms of the series are monotonically decreasing.
Now let's apply these conditions to each given series:
Series 1: The summation from n=1 to infinity of 1/n!
The terms of this series are positive since the factorial of any positive integer is positive. However, the terms of this series do not decrease since the factorial function grows rapidly with increasing values of n. Therefore, the integral test cannot be used for this series.
Series 2: The summation from n=1 to infinity of 1/n
The terms of this series are positive as the reciprocals of positive integers are positive. Moreover, the terms of this series do indeed decrease because as n increases, the denominator of each term becomes larger. Therefore, the integral test can be used for this series.
Series 3: The summation from n=2 to infinity of ln(n)/n^2
Similar to the previous series, the terms of this series are positive and the terms also decrease as n increases because ln(n) grows slower than n^2. Hence, the integral test can be used for this series.
Given the above analysis, it can be concluded that the series that could be tested for convergence/divergence using the integral test are II and III. Thus, the correct answer is c) II and III.