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Suppose $ \sum a_n $ and $ \sum b_n $ are series with positive terms and $ \sum b_n $ is known to be convergent.

(a) If $ a_n > b_n $ for all $ n, $ what can you say about $ \sum a_n? $ Why?

(b) If $ a_n < b_n $ for all $ n, $ what can you say about $ \sum a_n? $ Why?

a. We cannot say anything

b.If $a_{n}<b_{n}$ for all $n,$ then $\sum a_{n}$ is convergent

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Missouri State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

So now I have to Siri's a and and be in with public terms and beings known to convergent Let's say up Cut Toby in is Siri bian. Wait the index and some over net, remember? And so thus a to simply five notations. Okay, so first, what can I say about eight? And if a and it's going to be in for each term, Well, we cannot say Ah, ay least diverge aids divergent or convergence. There's no theorem to support us, and I can show you something. Demos The C A B in equals one. Oh, elsewhere. So this is P series record lectures in the book, sir. Um, let's say in equal so to Owen Square. So now a squid Ambien, And to be in this convergence and and it's also converted. But if we set up a and that's one or in and we know that the hark Siri's is gonna diverge, Okay, so I should be We know that be in is finite and for each term is less than being so definitely is gonna be finite. So a and worries oh is less than actually here with you, right? Be in instead of evenly. Okay,

University of Illinois at Urbana-Champaign