Determining Convergence or Divergence
Which of the serie...

Question

Determining Convergence or Divergence

Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.

∑ (from n=1 to ∞) (2ⁿ + 3ⁿ) / (3ⁿ + 4ⁿ)

Accepted Answer

Determine whether the series n equals 1 to infinity of 3 n + 4 n divided by 4 n + 5 n converges or diverges. We have two possible answers, being converging or diverging. Now, to solve this, we'll make use of the direct comparison test. This just tells us if we have a series, AN and BN, and A is between 0 and BN for all N. And be subn converges. A subn will also converge. Conversely, If we have the same series, where A is between 0 and B for all N, and A diverges, BN will diverge. Now, let's actually denote what our AN is. In this case, we'll say AN equals our series 3 n + 4 n, divided by 4 n + 5 n. Now we can actually make an estimate of this. We can say 4 in being greater than equals 1. Our numerator Simplifies to be 2 multiplied by 4 to the n. Because these would approach the largest base. So 3 end would approach 4 end, meaning we can just replace it. We do the same for the denominator. We have 4 end plus 5 end. This just approaches 5 to the end, and so we will put that in the denominator. So our comparison series then will be 2 multiplied by 4 to the end divided by 5 to the end. Which just simplifies even further to 2 multiplied by 4/5 all rays to the n. We'll call this B_N. Now we know that this converges already. Because this is geometric, where r or the ratio is the 4/5, which is less than 1. So based on our direct comparison tests, we know that A is less than B. And B converges, A will also converge. That means we can then say a subn. Converges Which means the answer to our problem is answer A. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Determining Convergence or DivergenceWhich of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.∑ (from n=1 to ∞) (2ⁿ + 3ⁿ) / (3ⁿ + 4ⁿ)

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Determining Convergence or Divergence
Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.
∑ (from n=1 to ∞) (2ⁿ + 3ⁿ) / (3ⁿ + 4ⁿ)