Problem 69 Hard Difficulty

For what values of $ r $ is the sequence $ \left\{ nr^n \right\} $ convergent?

Answer

Watch More Solved Questions in Chapter 11

Video Transcript

for what values of our is the sequence where the end of term is given by a M equals and are to the empower. We like to know when this is comm urgent. Let's look at two cases here case one absolute value are bigger than our equal to one. Then let's look at the absolute value of a n The answer. This is absolute value and our end Now I could use the properties of the absolute value to rewrite this. And I know this is bigger than her equal to the absolute value. And this is due to the fact that the absolute value our is bigger than our equals one. So officials at the limit absolute value and our end as n goes to infinity goes to infinity because it's bigger than the absolute value. And so really well, we should write. Is this bigger? They're equal to the limit and goes to infinity Absolute value n and that's infinity. So we have three cases, so we do new sense here. Therefore, we either have that the limit of a N is infinity negative infinity or does not exist and it won't exist because it might get larger and absolute value goes to infinity, but the signed by alternate. In that case, it won't diverged to infinity or minus eternity. The limit just won't exist. So on either of these three cases, by these three cases, I mean case one case too. And then Case three does not exist. We have that the sequence and are the end does not converge, so this will not be part of our answer. So we do not want to consider these values of our and absolute value that are larger than or equal to one. Because in that case, the sequence will not converge. We'LL go on to the next page to consider one last case These air the remaining values of our and we'LL show that it actually does converge in this case case too. Absolute value are less than one Now here, let's look at the limit and goes to infinity a m we'Ll show that this limit conversions by just evaluating it. Now let me rewrite this in two ways. First, let me replace the end with the ex and you'LL see why in a second why I'm doing this and rewrite the r by putting in the denominator. But changing the sign of the exponents now is X goes to infinity. This term goes to infinity and this term which equals one over r to the ex. I will also go to infinity since one over r is bigger than one an absolute value. So here we should use low Patel's rule but numerator and denominator or getting large and absolute value. So here I'll put the indicator to let us know that we're using Low Patel. This's a indeterminant form of type infinity over infinity. And that's why we're using well, Patel. So you go ahead and take the derivative of that numerator. That's just the one. And that in the denominator, this is exponential function. So we have natural log a bar, and we can rewrite this now as negative are to the ex over Ellen R. And this limit will be zero sense. R is between negative one and one and when you raise a number between negative one one to a limit and they'LL take the limit to infinity that always go to zero c, we have zero over Ellen are and that's still zero. So then the denominator noticed that this is defined. Except in the case when we have that are a zero so far zero We should treat that separately. So maybe here this is important because here natural log of zeros undefined. So here we need to throw in one more step. Our is not equal to zero but are equal zero case three is simple. So in case too absolute values less than one ours not zero. We showed that the sequence convergence to zero So conversions in this case now we'LL just fill in that hole at zero and show that it converges. This is the easiest part of the proof our equals zero then a n peoples and time zero to the end equals zero and this conversions to zero So converges therefore to summarize converges if and only if absolute value are is less than one And that's our final answer