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Numerade Educator

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Problem 42 Medium Difficulty

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \ln (n + 1) - \ln n $

Answer

Converges to 0

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Video Transcript

one thing that comment on is that if we just limit as n goes to infinity of a end and we tried to just plug in our inn equals infinity. Here we'd get infinity minus infinity. So we're definitely not allowed to do that from the minus. Infinity is indeterminate form. So we have to do something else just because plugging in and equals and Fendi gives us this indeterminate form that doesn't necessarily mean that the limit doesn't exist. It just means that we're not allowed to just plug in the value like that. So another thing that we could D'Oh, instead of just plugging in and equals infinity is toe use properties of the log function. So remember, subtraction corresponds to division with logs. So this turns into natural log of one plus one. Divided by n and the natural log function is a continuous function, which means that we're allowed to pull the limit inside of the function and then this limit should look should be pretty clear. But if this limit isn't clearly could do low petal, you could also divide top and bottom by n as in goes to infinity. That one over in is going to go to zero and you just get natural log of one other one. The natural log of one and natural log of one is zero. So our sequence does converge and it converges to zero.