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Determine whether the sequence converges or diverges. If it converges, find the limit If it diverges write NONE_ an = 4 - (0.4)" lim an 0300...

Question

Determine whether the sequence converges or diverges. If it converges, find the limit If it diverges write NONE_ an = 4 - (0.4)" lim an 0300

Determine whether the sequence converges or diverges. If it converges, find the limit If it diverges write NONE_ an = 4 - (0.4)" lim an 0300



Answers

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_{n}=\frac{n^{4}}{n^{3}-2 n}$

To figure out whether or not the sequence converges or diverges. I need to look at the limit as n goes to infinity of AM and figure out if that's the women exists. And if it's a finite number so we can do the trick where we look at the biggest power of in and the denominator and divide the top in the bottom, I at number so we can divide the top on the bottom by in cubed, and we have been on top than one minus two over and squared in the denominator. And as long as you don't get something an indeterminate form, you're allowed toe put the limit on top and the limit on the bottom. So as long as you don't get something like infinity over infinity or something divided by zero, then your levity. This here. We're just going to get infinity over one, so that's fine. That's not indeterminant form. It's just infinity. The limit does exist, but infinity is not a real number, so the limit didn't go to anything finite. So we say that the sequence diverges

In this problem, we see a sequence that oscillates between two values values mean zero and one just here for us. Yeah. Then two times 0, prepared again, three times zero and head again and so on. What we observe in this sequence is that The terms keep on oscillating between zero and 1, which means that A. N. Does not approach any particular number, which means that limit limit and tends to infinity of that sequence does not exist. This does not exist, which means that the sequence is diverse. That's

Sequence written in the problem? Yes. 2- for Brian one. Your car. As any in thesis we know there is a vehicle for by angle decrease. This is reciprocal and as any increases The relative to -4 billion will also increase Yeah. Because of negative side. All right. So Let's see as of one And should not be called to zero. Yeah, 2 -4 x one which is -2 As of two years to minus four by two Which is to -120 As of three years do minus four by three. Which is to wait three and so on. As infinity we can simply say limit And approaches infinity to -4 by hand. We just finally to we can say increasing sequence it clearly. So number one increasing sequence. Yeah. Yeah. And # two. Because of this value you can see it bounded. Mhm upper bounded and As of one swingers lower bounded. So increasing sequence and upper bounded. Mhm. These both concludes sequences and watching. Mhm. If we want to draw the back of the particular sequence, It will start from -2 -2 right And the value of one and then it will go on. has been taught with the value of two, so this is the graph of 2 -4 by and right.

The sequence that is soon here is three and let's go on a point. Hold it in. So just we can also do I get a three by four? Power and bless one by fourth. Power And right as both three x 4 and one x 4 is less than one. So we can simply say the limit and approaches infinity This value tree before our end plus one based on our end with approaches zero, you can which is zero. So as infinity is zero. And the sequences first few terms are as zero as pay power zero plus one upon four. Power zero, which is to as oneness three Power one last one upon for power one, which is four x 4, which is one and so on. So we can see the graph is starting from two and the value starts decreasing. So this is that joining function and this is the card. So sequence is converging for me. Okay. And limit. We have finally decided also limited infinity. Three power and plus one on four. Power end is zero.


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