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Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?$$\sum_{n=1}^{\infty} \frac{1}{\ln (n+1)}$$

convergent.

Calculus 2 / BC

Chapter 8

SERIES

Section 2

Series

Sequences

Missouri State University

Baylor University

Boston College

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Welcome back to New Murad. My name is Kevin Chirac. Let's take a look at the infinite Siri's um, NC denoted by the summation from n equals one to infinity of one over the natural log of N Plus One. And we're looking for whether or not this serious converges and the way that we can tell is just by simply listing a few of these terms out. So we're going to generate this. In other words, we're going Teoh, step each of these and values up along the counting numbers so we'll start with an equal to one because we have that notated here. So be one plus one. Well, then add that to one over the natural log of two plus one, and we'll continue this process until we were to get Teoh the eighth term one over the natural log of a plus one. We wouldn't want to take a look at the partial summation, so the partial summations would be what happens when we add just the first, however many terms. So for the 1st 1 for example, be one divided by the natural log of to, and that would get us one point for four Teoh six. Looks like 95 will include that one for the second partial summation, we would include up till the second term. So now we would add an additional natural logger one over the natural log of three and they'll get us a total of two point 35 to nine 34 And that would be our second. A partial summation. Now we could do this one more time for S three and s four. So I understand a calculate that's really quickly. So now we're just going Teoh, do it for the natural log of four and that's going to get 3.7 for 28 Looks like to A to we could then do it for the fourth. And that would get us three point 69 561 I think we're showing promise. I think we might have locked in this digit. Now I don't know that we're gonna grow much above three. Let's see, we add it. Teoh. The natural, the reciprocal. The natural log of six? Nope. Nope. They blew right past it. So now we're at the harsh estimation. Up till five would be 4.25 37 to 7. Partial the partial summation up to six and all I'm doing is just typing this in. My calculators were working through together, so I got 4.767 6 to 5. Looks like seven. Let's do two more, so the next one is going to be plus one over. The natural log of eight gets 5.2485 to 4038 And I'm going to say the last one for you because what I want to point out with the last few minutes this video is that what we would be expecting to have happened is similar to what I thought was happening in the green here. Some of these digits would begin toe lock in, and they wouldn't continue to step up. But that doesn't seem to be the pattern. This four continues to grow onto the five and all of the other future digits air kind of following along with it. So this seems like it is continuing to grow, and it's continuing to grow in such a way that it is not approaching a visible living. So I am going to claim that from the 1st 8 terms. This sequence does not. The Siri's does not imply that it is convergent. So I would conclude that his divergent based on the 1st 8 terms of the partial summation Great. I hope this helped out if it did take him.

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