Determine whether the series ? (n=1 to ?) ln((n+3)/(n+4))...

Determine whether the series ? (n=1 to ?) ln((n+3)/(n+4)) converges or diverges. If it converges, find its sum. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The series converges because the sequence of partial sums {sn} converges. The sum of the series is . (Type an exact answer.) B. The series diverges because it is a geometric series with |r| ? 1. C. The series converges because it is a geometric series with |r| < 1. The sum of the series is . (Type an exact answer.) D. The series diverges because the sequence of partial sums {sn} does not converge.

Transcript

00:01 Hi, here we have an infinite series with terms natural logarithm of n plus 3 over n plus 4, n is going from 1 to infinity and we want to figure out if it converts or diverges.

00:14 We have a list of possible answers and we need to figure out which one is the correct one.

00:21 The correct one actually is that the correct answer from the ones you have available is that the series diverges.

00:32 Because the sequence sn of the partial fractions does not convert.

00:40 Okay, how we are going to show this? let's see why this is the case.

00:46 So let's construct them.

00:51 So the answer is this one, that it diverges because the sequence of the partial sums does not converge.

00:59 So this is the answer and now we are going to so why this is the case.

01:05 All right so take now s n the nth partial sum.

01:10 So this is going to be equal to the sum k from one to n of the natural loga root natural logarith n plus 3 over n plus 4.

01:23 And now let's see how this looks like if we take uh oops error here.

01:29 So we need to hear k plus 3 over again.

01:32 Pass law 4.

01:33 So let's write down this, expand this sum.

01:36 So for k equal to 1 we have the natural logarithm of 1 plus 3 minus the natural logarithm of 1 plus 4.

01:46 This is for k equal to 1 plus 4.

01:49 4k equal 2 we have the natural logarithm of 2 plus 3 minus the natural logarithm of 2 plus 4.

01:59 This is for k equal 4.

02:02 Plus for k equals 3 we have 3 natural logarithm of 3 plus 3 minus the natural logarithm of 3 plus 4 this is 4k equals 3 then let's do one more natural logarith for k equal 4 we have 4 plus 3 minus the natural logarithm of 4 plus let me see this is 1, k equals 1, k equals 2...

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