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JH
Numerade Educator

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Problem 2 Easy Difficulty

Explain what it means to say that $ \sum_{n = 1}^{\infty} a_n = 5. $

Answer

The sum of all the terms is 5 or the limit of its $n$ th partial sum $\rightarrow 5$ as
$n \rightarrow \infty .$

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Video Transcript

So for this problem, let's explain what it means to say that the sum from one to infinity of a N equals five. So what this is really saying is that we can go ahead and rewrite this left hand side is a limit. Let's go from the living as Kay goes to infinity as we saw it from any Poles want. Okay, so is he like a go to infinity? This upper bound up here is getting closer to infinity and a N equals five. So really one way to explain what it means is it means that if you take this fight, I some let's go, let's go ahead and rewrite this Lim Kay goes to infinity. So now I'm just going to expand the sigma notation a one a two all the way up to a case. Okay, just fixed the arbitrary number. And that is we like, hey grow without bound that we end up with five. So that's basically what it means to say This next part is just the bonus fact. Here's a more precise way of saying every writing what the limit beats If you've seen the Absalon definition, the definition of death if you see the Absalon definition and the way that what this means is for every number Absalon bigger than zero, there exist some large number capital end such that if K is bigger than N, then we have that the distance between five and our approximation our finances is less than absolute. That's the more precise way to say that the living goes to five, and that's our final answer.