Problem 49 Easy Difficulty

Prove the Root Test. [Hint for part (i): Take any number $ r $ such that $ L < r < 1 $ and use the fact that there is an integer $ N $ such $ \sqrt [n]{\mid a_n \mid} < r $ whenever $ n \ge N.] $

Answer

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Video Transcript

in this question we're asked to prove the root test. So let's look at the first part and compare it to the proof of the ratio test here we have the limit as an approaches infinity of the anthem route of the absolute value. A suburban. This equals a value L. Which is less than one. So we can find a number are that's in between L. And one. This allows us to say that there's in N. Thank you. So that this N fruit is less than our for and that's bigger than or equal to this end that we've chosen. Okay Now we can rewrite this inequality as the absolute value of a seven is less than our to the end and we can change this end value. So we have. Okay the absolute value of a sub N minus or plus one. Let me make that a little bit clearer for. Yeah This will be less than our to the end plus one Because we're just replacing the end with the end plus one. So if we were to take a look at this series starting with K equals N to infinity absolute value. Ace. Okay muse equals the value of a sub. And plus the absolute value of a sub and plus one on and on and on forever. We can say that this would be less than or to the end plus R. to the n. plus one. On and on forever. Now these are Zara geometric series with Are less than one. So it's convergent and because our original series is less than a convergent series. This series converges by the comparison test. The first end terms don't matter because they will not affect the convergence or divers. It just matters what happens um from and to infinity. Okay, that's the first part. The second part is the divergence criteria. So if we have the limit as N approaches infinity of the and fruit of a sub N and we say this is equal to L. If L is bigger than one, then again we can find a value are that's in between one and L. So we can say the absolute the N. Through of the absolute value is greater than or Which is greater than one. And so the absolute value is greater than or to the end which is greater than one, raise the young, which is just one. But this means that the limit as and approaches infinity uh is so Ben Is not equal to zero. So this series diverges by the divergence test, you need that limit to equal zero. Uh and if it doesn't then the series can't converge. Okay, let's take a look at the last part. The last part is when it's inconclusive, If the limit is equal to one. So this means that you can't tell whether it converges or diverges similar to the ratio test and you can use the same example, we have the some of one over N. In the some of one over and squared. We know that this diverges this converges. Yeah, but both have the limit as an approaches infinity of this, and route shows that the root test is inconclusive When the limit is equal to one.