Which of the following is true about the Comparison Test?
I. Terms can oscillate.
II. If the "bigger" series diverges, then the "smaller" series diverges.
III. If the "smaller" series diverges, then the "bigger" series diverges.
a) I only
b) II only
c) III only
d) II and III only
read up on the test.
all terms must be positive
clearly III is true
II is not necessarily true. The smaller series may or may not diverge.
To determine which of the statements are true about the Comparison Test, let's first understand what the Comparison Test is. The Comparison Test is used to determine the convergence or divergence of a series by comparing it to another series that is known to converge or diverge.
I. Terms can oscillate:
This statement is not true about the Comparison Test. The Comparison Test does not consider the oscillation of terms; it only compares the size of the terms in two series.
II. If the "bigger" series diverges, then the "smaller" series diverges:
This statement is true. If the terms of a series are always smaller than the terms of a divergent series, then the smaller series must also diverge. This is because if the larger series, which is known to diverge, can't be contained, then the smaller series won't be able to either.
III. If the "smaller" series diverges, then the "bigger" series diverges:
This statement is also true. If the terms of a series are always greater than the terms of a divergent series, then the larger series must also diverge. This is because if the smaller series, which is known to diverge, surpasses a certain threshold, the larger series will definitely surpass that threshold as well.
Considering the explanations above, the correct answer is:
d) II and III only