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convergence_intuition

convergence_intuition - Intuition for the convergence of...

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Intuition for the convergence of certain types of series Math 8 2009, Chris Lyons Let’s start with a few examples: Example 1. Look at the series k =1 1 k 3 + 4 k - 1 . (1) Does this converge or diverge? Discussion. Obviously this series (1) is not the same thing as k =1 1 k 3 . (2) If it were, this would be easy, because we know that (2) converges! However, (1) kind of looks like (2), in the following imprecise sense: using to mean “approximately”, if k is very very large, we have k 3 + 4 k - 1 k 3 , just because k 3 grows so much faster than 4 k - 1 in the expression k 3 + 4 k - 1. So this means that we ought to have 1 k 3 + 4 k - 1 1 k 3 . Using this imprecise reasoning, we would guess that, since (2) converges, then so does (1). Now here’s the more precise reasoning for why (1) should converge: because all terms of the series are positive and because lim k →∞ 1 k 3 + 4 k - 1 / 1 k 3 = lim k →∞ k 3 k 3 + 4 k - 1 = 1 (as you should be able to show!), the limit comparison test says that (1) converges because (2) converges. Example 2. Look at the series k =1 k 2 + 3 7 k 6 + 18 k 5 - 1 . (3) Does this converge or diverge?
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