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SACOLUJ21192 ~Discussion section 5Jopic#Zan and Zb both diverge: Ihen L(o+26,) converges Select one: a. True b. False...

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SACOLUJ21192 ~Discussion section 5Jopic#Zan and Zb both diverge: Ihen L(o+26,) converges Select one: a. True b. False

SACOLUJ21192 ~Discussion section 5 Jopic #Zan and Zb both diverge: Ihen L(o+26,) converges Select one: a. True b. False



Answers

Decide if the statements are true or false. Give an explanation for your answer. If $a_{n}>0.5 b_{n}>0$ for all $n$ and $\sum b_{n}$ diverges, then $\sum a_{n}$ diverges.

So if a then and be then both diverge, then we want to figure out if the event plus the event will also diverge. So let's let's just look at an example here where event is equal to x squared. Um and we'll say it's just X squared plus one and then be a then is equal to negative X square. So if we have these two sequences obviously in, sorry, I should have made this, not using my X variable with my invariable and so obviously both of these sequences diverge. Um and squared plus one is going to go to infinity and negative and squared is going to go to negative infinity. But if we add these two together, so if we have and squared plus one and we add negative and squared, this is going to be equal to what we can cancel these end squares and it's just going to be equal to one at all values of N. And that means that obviously this sequence would divert or would converge. So this is a false statement. If a an and BN diverged, then it is not necessarily true that a N plus BN will diverge

Hello. So if the sequence a cement converges to three and B seven converges to to, that means the limit as N approaches infinity of a sub N is equal to three. And the limit as N approaches infinity of B seven is equal to two. So then we have the limit as an approaches infinity of while a seven plus B sub N. That's equal to the limit as N approaches infinity of a seven plus. The limit as N approaches infinity of pizza is limited and approaches infinity of a suburban plus the limit as an approaches infinity of peace, a man or that's going to be equal to just three plus two which is equal to five. So therefore, yes, the Um the sequence a 7-plus piece of end does converge to five. So therefore the given statement is indeed true.

The statement here is if a convergent sequence has us and Less than or equal to five or all end, then the limit as n approaches infinity. For the sequence must be less than or equal to fight. And this is a true statement. Okay? Um this could be proven with an absolute delta proof, but um I don't think this has been covered yet. So um the explanation would be um if The limit was greater than five, then you would have to have Um you would have to have terms that are greater than five. Right? Because the thing is, the thing with limit is that if you're approaching a number that's greater than five, then it has to lead up there. So which means you would have terms that are greater than five, but that's not true because all the terms here are less than or equal to fight, so therefore this is a true statement.

Number 31 possible do conditions absolutely convergent or conditional conversion. So, yes, doctors actually a true statement because if a particular sees, for example, it's it's a in have bean n from one to infinity, culpable this. Then when we deliver lapse, your condition conversion than this feet off about the cities, the cities should be conversion and then we talk about its absolute value. This city's should also be convergent. Then we'll say, absolutely conversion. And if the series is divergent, then we'll say they are these. The city's has condition you can so the strange ministry


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