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Numerade Educator



Problem 87 Hard Difficulty

Suppose that a series $ \sum a_n $ has positive terms and its partial sums $ s_n $ satisfy the inequality $ s_n \le 1000 $ for all $ n. $ Explain why $ \sum a_n $ must be convergent.




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Video Transcript

suppose that this Siri's has positive terms. So here we're just supposing that a M is bigger than zero for all in, and we're given inequality for the partial. Some sn recall I send is the sum of the A's all the way up to and in this case we don't know what the starting point is, because the signal notation, it doesn't tell us where to start, but it doesn't matter what the starting point is. So if we were to just start at one, we would add up all the A's until we get to a M. That's the end partial some. And we're told that no matter how big is this thing is always less than one thousand. And now we'd like to explain why this some conversions? Well, first of all, since and it's bigger than zero for all in What can we say about the sequence? Sn, Is it increasing or decreasing? I claim it's increasing. So since a M is bigger than zero for all and s n plus, one is bigger than s n. And the way to see this is just the subtract. So if we subtract, he's using this definition of here we'Ll end up with a and plus one So maybe I could actually show that Let me go a few steps back here So this is a one all the way up to a end and then a n plus one That's s m plus one there. And then over here we have a miner's a one all the way up to an and then when you subtract, you could cancel everything from a one all the way to a m. But your leftover with this a m plus one and by assumption, all the ends or positive. So this must be a positive number. This means that s n plus one is bigger than s n. You get that by just adding s end to both sides of this inequality. So this means that SN is founded. The reason is bounded is not due to what we should show. The reason is bounded is due to this upper bound that's given and increasing, which is also called Mama Zone. So the SN converges in other words, this thing right here conversions by this is firm section eleven point one and there's a name for the serum. This is called. Let me find the name of the serum. Believe it's the monotone sequence there. Um, this is from eleven point one yes. Monotone sequence. Tehran which says that any bounded monitor in sequence must converge. Therefore, this conversions. So that's our final answer.