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Numerade Educator

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Problem 30 Easy Difficulty

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \sqrt { \frac {1 + 4n^2}{1 + n^2}} $

Answer

converges to 2

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Video Transcript

for us to determine whether or not the sequence converges. There diverges. We need to look at the limit is n goes to infinity of a M And if this limit exists in his finite than the sequence is said to converge, otherwise that is said to diverge. Square root function is a continuous function, which means that we're allowed to pull the limit inside of the square root. Okay, with continuous functions, you can pull the limit inside of the functions. That's what we're doing here and now we could do the trick where we look at the term that's going to infinity, the fastest in the denominator, and then divide the numerator and the denominator by whatever that is. So in this case, we'LL divide the numerator and the denominator by in squared says that we have on top now and then at the bottom. We have went over and squared plus one as in goes to infinity. This and this are both going to go to zero and we're gonna be left with square root of four over one, which is the square to four, which is too. So we converge two, two