If the summation of a sub n and the summation of b sub n are both series of positive terms and an < bn for all n, and the summation of b sub n diverges, then (15 points)
A) the summation of a sub n converges
B) the summation of a sub n diverges
C) the summation of a sub n = 0
D) the behavior of the summation of a sub n cannot be determined
D
Consider b_n = 1
a_n = 1/n^2
or
a_n = 1/n
∑1 surely diverges, but 1/n^2 converges, and 1/n diverges.
To determine the behavior of the series of a sub n, given that the series of b sub n is divergent and an < bn for all n, we can use the comparison test.
The comparison test states that if we have two series, a sub n and b sub n, where 0 ≤ a sub n ≤ b sub n for all n, and the series of b sub n converges, then the series of a sub n also converges. If the series of b sub n is divergent, then the behavior of the series of a sub n cannot be determined.
In this case, we know that an < bn for all n and the series of b sub n is divergent. This situation falls into the second case of the comparison test, where the series of b sub n is divergent. Therefore, the behavior of the series of a sub n cannot be determined.
Hence, the correct answer is:
D) the behavior of the summation of a sub n cannot be determined.