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Numerade Educator

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Problem 60 Hard Difficulty

Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)
$ a_n = \sqrt[n]{3^n + 5^n} $

Answer

$$
5, \text { we take } \lim _{n \rightarrow \infty} a_{n} : \lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} \sqrt[n]{3^{n}+5^{n}} \stackrel{(1)}{=} \lim _{n \rightarrow \infty} \sqrt[n]{5^{n}}=5
$$

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Video Transcript

okay for this problem when our graphing it we just want a plug in different values of n assuming that we're just doing this by hand. So for equals one, we just have three plus five, which is eight case when in his one we're up here at eight. And then if you want to get a better graph, then you just plug in more points. And when you do this, your graph is going to start to look like this. You see that? It looks like we're we have a horizontal Assam Toto here at five. So it looks. Looks like we converge two five is Remember, when you're trying to figure out exactly what your sequence converges to by looking at a Graff, you want to keep an eye out for these horizontal aston totes. So as you go further to the right, what does it look like? Your functions approaching. So as we go further to the right here, it looks as though our function is approaching five. Okay, so if we wanted to prove this, what we could do is we could use the use the trick where we look for the largest term that's happening by largest, we mean the thing that's going to infinity the fastest. So it appeared that would be five to the end. So look at the thing that's going to infinity, the fastest. And then we can factor that out. So pull out this five to the end and then we'd have ah, three over five to the end here, plus one right. And we're taking the square rule or the end through that's multiplication function. And by that we mean that it's we can write it as the in through tw of five to the end, multiplied by the in through tw of you know, the other stuff three over five to the end, plus one. So the in through is really just taking whatever's in here to the one over in tower. So if we take five to the end to the one over in power, you should get five, okay. And then over here we can pull this limit inside. And once we do that, this term is going to go to zero because three or five others less than one an absolute value. OK, so like we mentioned, this term goes a zero and then we start this one here. So I guess I should still have this limit. So it turned out to be five times the in through have won. But any root of one is still going to be one. So it all simplifies to just five. So the sequence converges two five.