Assume that bₙ is a sequence of positive numbers converging to...

Question

Assume that bₙ is a sequence of positive numbers converging to 4/5. Determine if the following series converge or diverge.

b. ∑ (from n = 1 to ∞) (5/4)ⁿ (bₙ)

Accepted Answer

Let a subn be a sequence of positive numbers such that a subn approaches 3 halves as n approaches infinity. Determine whether the series sum from n equals 1 to infinity of 2/3 raised to the n, a subn converges or diverges. And of course we have two possible answers being converges or diverges. Now, we'll make use of the ratio test to solve this, which tells us for a series, a subn. And that's greater than 0, we have l, which is the limit asn approaches infinity. Of AN + 1 divided by AN. If L is less than 1, the series converges absolutely. If L is greater than 1, the series diverges, and if L equals 1, the test is inconclusive, and we have to choose a different test. So, let's go ahead and set up our ratio here. We can then see that BN will just be 2/3 raised to the n. A subn. Now, to solve this. We will use our ratio test here. To use the ratio test, keep in mind that we can also use the absolute value. On our ratio for our series. So, we will write this. As the absolute value of bn + 1 divided by bn. Now we'll replace our N with N + 1, giving us 2/3, race to the N + 1, A N + 1. Divided by 2/3 raised to the n a subn, all in the absolute value. Now we can simplify the geometric part of this ratio, or in other words, the part with the 2/3. Because we have 2/3 raised to the n plus 1 divided by 2/3 raised to the n, by exponent rules our n will cancel. That gives us just 2/3. This is then multiplied still by a subn + 1 divided by a subn. Now, from here. Let's calculate our limit. We have L Equals the limit As N approaches infinity of 2/3 multiplied by a subn + 1 divided by a subn. So here, we have 2/3. So we can then say. 2/3 is multiplied by the limit as it approaches infinity. Of AN + 1 divided by AN. Now we're given that our limit of a subn equals 3 halves from our question. If this converges to a non-zero limit, then the shifted sequence also converges to that same number. So, in other words, the limit of AN + 1 and the limit of AN are the same value. We can then say L equals 2/3 multiplied by 3 halves divided by 3 halves, which is simplifies to give us 2/3. So L As 2/3, which is less than 1, which means this converges. That means the answer to our problem is answer A. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Assume that bₙ is a sequence of positive numbers converging to 4/5. Determine if the following series converge or diverge.b. ∑ (from n = 1 to ∞) (5/4)ⁿ (bₙ)

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Assume that bₙ is a sequence of positive numbers converging to 4/5. Determine if the following series converge or diverge.
b. ∑ (from n = 1 to ∞) (5/4)ⁿ (bₙ)