Determine if the following series converges, diverges, or is inconclusive. ∑ n = 1 ∞ ( 4 n + 1 n − 2 ) n \sum_{n=1}^{\infty}\left(\frac{4n+1}{n-2}\right)^{n}

this problem, we're asked to determine if the following series converges, diverges, or is inconclusive. Now how do we solve this problem? Well, I can see we have this sum from n equals one to infinity, and that's four n plus one over n minus two, all raised to a power of n. Since I have this case where we have this whole piece right here, a sub n raised to a power of n, that's a sign that I'm gonna want to use the root test to solve this problem. Now the root test works like this. You wanna calculate r, and r can be found by taking the limit as n approaches infinity for the nth root of the absolute value of a sub n to the nth power. So because of this, well, I can use this to solve my problem up here and apply this test. So to find r, that's gonna be the limit as n approaches infinity of the nth root of the absolute value of this whole piece right here. So we have four times n plus one over n minus two, all raised to a power of n. Now what I'm going to do from here is I'm going to see how this function behaves as n approaches infinity. Now initially, if we plug one in for n, because we start at one here, we will get a negative value that gets output. But as we go to two or three, as we keep going up in the long run, you're going to notice that this number is actually gonna become much larger than the negative two we have right there. So because of that, we can say that these values in the long run will be positive, meaning we can knock off these absolute value bars. So at this point, I can just go ahead and cancel this nth root and this n here, giving me the limit as n approaches infinity for four n minus one, or four n plus one, I mean, over n minus two. Now from here, what I want to do is see how what's gonna happen as n approaches infinity. Now for this case, I'm gonna look for the highest power of n that we have on the top and bottom of our fraction. You see the highest power is n to the first here, the highest power on bottom is n to the first there. So because we have the same exponents on top and bottom, I need to find a ratio of the coefficients out front. We can imagine there's a one in front of that n in the denominator, so because of that, this whole thing is going to come out to four over one, which I know is just four. So that's gonna be our r. So now that we have the r value, well, remember, this r tells us how we can determine whether our series converges, diverges, or is inconclusive. Because if we have a case where r is less than one, then we would say that our series converges, similar to the ratio test. If r is greater than one, then our series diverges. However, if r is equal to one, then we would say that our test is inconclusive. Now looking at this, I can see that we have an r value of four, and four, I can see, is clearly greater than one. So because of that, we can conclude this series diverges, and that would be the solution to our problem. See you in the next one.

14. Sequences & Series

Convergence Tests

Calculus

14. Sequences & Series

Convergence Tests

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Multiple Choice

Determine if the following series converges, diverges, or is inconclusive.
$\sum_{n = 1}^{\infty} {(\frac{4 n + 1}{n - 2})}^{n}$

Converges

Diverges

Inconclusive

Verified step by step guidance

Step 1: Recognize that the series involves a term raised to the power of n, which suggests that the root test or ratio test might be useful for determining convergence or divergence.

Step 2: Apply the root test. The root test involves evaluating $ ho = ext{lim}_{n \to \infty} \sqrt[n]{a_n} $, where $ a_n $ is the general term of the series. If $ ho < 1 $, the series converges; if $ ho > 1 $, the series diverges; if $ ho = 1 $, the test is inconclusive.

Step 3: Substitute $ a_n = \left(\frac{4n+1}{n-2}\right)^n $ into the root test formula. This simplifies to $ \rho = \text{lim}_{n \to \infty} \left(\frac{4n+1}{n-2}\right) $, since the $ n $-th root cancels the exponent $ n $.

Step 4: Simplify $ \frac{4n+1}{n-2} $ as $ n \to \infty $. Divide numerator and denominator by $ n $ to get $ \frac{4 + \frac{1}{n}}{1 - \frac{2}{n}} $. As $ n \to \infty $, $ \frac{1}{n} \to 0 $ and $ \frac{2}{n} \to 0 $, so $ \frac{4 + \frac{1}{n}}{1 - \frac{2}{n}} \to 4 $.

Step 5: Conclude based on the root test. Since $ \rho = 4 > 1 $, the series diverges.

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Struggling with Calculus?

Determine if the following series converges, diverges, or is inconclusive. ∑n=1∞(4n+1n−2)n\sum_{n=1}^{\infty}\left(\frac{4n+1}{n-2}\right)^{n}

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Convergence Tests practice set

Determine if the following series converges, diverges, or is inconclusive.
$\sum_{n = 1}^{\infty} {(\frac{4 n + 1}{n - 2})}^{n}$