Expii

# What Are the Series Comparison Tests? - Expii

The idea behind comparison tests is to compare a complicated series to a simpler one. The direct comparison test says that if $$(a_n),(b_n)$$ are two sequences of nonnegative numbers such that $$0\le a_n\le b_n$$ for all $$n$$, then: (1) if the series $$\sum b_n$$ converges, then so does the series $$\sum a_n$$; or put another way, (2) if the series $$\sum a_n$$ diverges, then so does $$\sum b_n$$. The limit comparison test says that if $$(a_n),(b_n)$$ are two sequences of positive numbers such that the limit $$\lim_{n\to\infty} \frac{a_n}{b_n}$$ exists and equals a finite nonzero (positive) number, then the two series $$\sum a_n,\sum b_n$$ either both converge or both diverge.