well, what does the Comparison Test actually say?
If we have two series ∑ a n and ∑ b n with a n , b n ≥ 0 for all n and a n ≤ b n for all n . Then, If ∑ b n is convergent then so is ∑ a n . If ∑ a n is divergent then so is ∑ b n .
So, what does this mean? I, II, I AND II only , OR I AND III only
The Comparison Test is a theorem in mathematics that helps determine the convergence or divergence of a series. It provides a way to compare the behavior of two series and draw conclusions about their convergence based on the behavior of one of the series.
The Comparison Test states that if we have two series, ∑aₙ and ∑bₙ, where aₙ and bₙ are non-negative for all terms n, and aₙ is always less than or equal to bₙ, then the convergence behavior of one series can be determined by the convergence behavior of the other series.
There are three possible conclusions we can make based on the Comparison Test:
1. If ∑bₙ converges, then ∑aₙ also converges. This means that if the series of bₙ terms converges (i.e., the sum of the terms approaches a finite value), then the series of aₙ terms also converges.
2. If ∑aₙ diverges, then ∑bₙ also diverges. This means that if the series of aₙ terms diverges (i.e., the sum of the terms does not approach a finite value), then the series of bₙ terms also diverges.
3. If ∑aₙ converges, or if ∑bₙ diverges, we cannot make a conclusion about the convergence or divergence of the other series. This means that if the series of aₙ terms converges or if the series of bₙ terms diverges, we cannot determine the convergence or divergence of the other series based on the Comparison Test alone.
Based on the information provided, the correct answer to the given options is "I and II only." This means that the Comparison Test tells us that if ∑bₙ is convergent, then ∑aₙ is also convergent (statement I), and if ∑aₙ is divergent, then ∑bₙ is also divergent (statement II).