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What is the Direct Comparison Test for Convergence of an Infinite Series?

Answer 1

If you are trying determine the conergence of #sum{a_n}#, then you can compare with #sum b_n# whose convergence is known.

If #0 leq a_n leq b_n# and #sum b_n# converges, then #sum a_n# also converges. If #a_n geq b_n geq 0# and #sum b_n# diverges, then #sum a_n# also diverges.

This test is very intuitive since all it is saying is that if the larger series comverges, then the smaller series also converges, and if the smaller series diverges, then the larger series diverges.

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Answer 2

Charles Anctil

The Direct Comparison Test for convergence of an infinite series states that if the terms of a series ( \sum_{n=1}^{\infty} a_n ) are all nonnegative and can be bounded from above by the terms of a convergent series ( \sum_{n=1}^{\infty} b_n ), then ( \sum_{n=1}^{\infty} a_n ) also converges. Conversely, if the terms of ( \sum_{n=1}^{\infty} a_n ) are all nonnegative and are bounded from below by the terms of a divergent series ( \sum_{n=1}^{\infty} b_n ), then ( \sum_{n=1}^{\infty} a_n ) also diverges.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.