Determine the convergence or divergence of the series. ∑ k = 1 ∞ 7 k 2 5 k + 3 \sum_{k=1}^{\infty}\frac{7k^2}{5k+3}

in this problem, we're asked to determine the convergence or divergence of the following series. Now here we have the sum from k equals one to infinity of seven ks squared over five ks plus three. Notice we're not told what type of series we're dealing with. We're not told what type of test to use, and I also didn't provide the flowchart in this video. I encourage you to try solving this problem yourself and see if you can do it without using the flowchart. However, if you find yourself getting stuck at any point, feel free to go ahead and take a look at the flowchart and see how you can go about solving this. So let's go ahead and jump right in. Well, right off the bat, I'm not seeing any kind of familiar forms or anything that's jumping out to me when looking at this series. So what I'm just gonna go ahead and do is start with the divergence test, which is always a safe place to start. So we're gonna take the limit as n approaches infinity for a sub n. Now a sub n is going to be this piece right about here. Now, what I can do is take a look and see how this is going to behave. And there is something that I noticed just looking at this off the bat. I noticed that our highest exponent on k that we have right here is k squared. And notice that we also have this k in the denominator, but that's k to the first power. And also, since we're dealing with k here, we should really change this a sub n to a sub k. So we have a sub k that we're dealing with, and this is actually going to be k that's approaching infinity, and this is going to be a sub k. So that's something to keep in mind. Just notice the variable switch there, but we can still solve this in the same way. So now that I can see that k here is approaching infinity, and the higher exponent of k shows up on top, well, when that happens, I I can see that that means this whole fraction is just going to blow up to either positive or negative infinity. Since I can clearly see here that there's nothing on in the numerator that's gonna cause this to become negative, and this isn't some sort of negative piece that we're dealing with here, then that means that this is going to go to positive infinity. So we would say that this limit here is just gonna blow up the positive infinity, and notice that positive infinity clearly does not equal zero. So because we got a case that does not equal zero, by the divergence test, our series diverges. That's how the divergence test works. It is either going to immediately diverge, or if not, it's gonna become inconclusive. But we can't we can't test to see if it's convergent, but we can see that it diverges, and that was shown right here. So that is how we can use the divergence test to solve this problem, and that is how you can deal with these series where you're not sure what test to use. It's always safe to start with the divergence test and then move on from there. Hope you found this helpful.

14. Sequences & Series

Convergence Tests

Calculus

14. Sequences & Series

Convergence Tests

Struggling with Calculus?

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Multiple Choice

Determine the convergence or divergence of the series.
$\sum_{k = 1}^{\infty} \frac{7 k^{2}}{5 k + 3}$

Inconclusive

Diverges

Converges

Verified step by step guidance

Step 1: Recognize that the problem involves determining the convergence or divergence of the infinite series $ \sum_{k=1}^{\infty} \frac{7k^2}{5k+3} $. To analyze this, we can use standard tests for convergence such as the Comparison Test, Limit Comparison Test, or the Ratio Test.

Step 2: Observe the general term $ \frac{7k^2}{5k+3} $. As $ k \to \infty $, the dominant term in the numerator is $ 7k^2 $ and in the denominator is $ 5k $. Simplify the fraction for large $ k $ to approximate $ \frac{7k^2}{5k+3} \approx \frac{7k^2}{5k} = \frac{7k}{5} $. This suggests the series behaves similarly to $ \sum_{k=1}^{\infty} k $.

Step 3: Recall that the series $ \sum_{k=1}^{\infty} k $ is a divergent series because the terms do not approach zero and the sum grows without bound. This provides an initial indication that $ \sum_{k=1}^{\infty} \frac{7k^2}{5k+3} $ may also diverge.

Step 4: Apply the Comparison Test or Limit Comparison Test to confirm divergence. Compare $ \frac{7k^2}{5k+3} $ with $ k $. Compute the limit $ \lim_{k \to \infty} \frac{\frac{7k^2}{5k+3}}{k} $. Simplify the expression to find $ \lim_{k \to \infty} \frac{7k^2}{k(5k+3)} = \lim_{k \to \infty} \frac{7k}{5k+3} $. As $ k \to \infty $, this limit approaches $ \frac{7}{5} $, which is a positive finite constant.

Step 5: Conclude that since $ \frac{7k^2}{5k+3} $ behaves similarly to $ k $ and $ \sum_{k=1}^{\infty} k $ diverges, the original series $ \sum_{k=1}^{\infty} \frac{7k^2}{5k+3} $ also diverges by the Limit Comparison Test.