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Numerade Educator

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Problem 32 Easy Difficulty

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \cos \left( \frac {n \pi}{n + 1} \right) $

Answer

converges to $-1$

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Video Transcript

to find out whether or not this sequence converges. There diverges. We look at limited n goes to infinity of an If this limit exists and is finally then our sequence is said to converge. Otherwise they're sent to diverge. Since the cosign function is a continuous function were allowed to pull the limit inside of the coastline function. Now we do the trick where we look at the denominator. Look at the term that's going to infinity, the fastest in the denominator, and then divide the top and the bottom by whatever that is. So this case will divide the top and the bottom by N and I will have tie up top and one plus one over in in the denominator. As n goes to infinity, one over in is going to go to zero and we're gonna be left with co sign of pie. Coastline of pi is minus one. So we converge two minus one