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 ∞) sin (1/n)

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine whether the series, the sum evaluated from n equals 1 to infinity of tangent of 1 divided by n converges or diverges. OK, so it appears for this particular prompt we are given a series and were asked to determine whether or not this particular series will either a converges or b diverges. So now that we know that we're ultimately trying to determine whether or not this series will converge or diverge, we need to recall and note that 4 n is greater than or equal to 1, the terms. Tangent of 1 divided by n greater than 0. And the terms tangent of 1 divided by n as it approaches 0. As n approaches infinity, and we need to note that based on these terms, therefore, thus the necessary vanishing condition holds, but it is not sufficient enough to prove convergence. So because it's not sufficient for proving convergence, we must now recall and use the small angle limit. So we need to note that if we let, so let's make a note here. So if we let. X approach 0. It is standard that we can go ahead and write that the limit as x approaches 0 of tangent of x divided by x is equal to 1, and if we set x is equal to 1 divided by n, this will give us. The limit as n approaches infinity of tangent of 1 divided by n divided by 1 divided by n. OK, and this will be equal to 1. So now, at this stage, we need to recall and use the limit comparison test. And as we should recall based on our prior knowledge, and you should be able to look this up in your textbook or quickly online in case you forget, but the limit comparison test states that if we let a n be greater than 0 and bn be greater than 0, for all lowercase n is greater than or equal to N, where N is an integer value. And if we assume the limit L is equal to the limit as n approaches infinity of a n divided by bn exists in square bracket of 0, infinity parenthesis. And if 0 is less than L and L is less than infinity, then the sum of a n and the sum of bn either both converge or both diverge. And if l is equal to 0 and the sum of bn converges, then the sum of a n converges, and if the limit is equal to infinity or if l is equal to infinity and the sum of b n diverges, then the sum of a n diverges. Otherwise, the test will be inconclusive, meaning we will need to use another test. So with that said, we now need to apply our limit comparison test with our harmonic series. The sum. Of 1 divided by n, and we need to note that since the limit as n approaches infinity, specifically lowercase n approaches infinity, of tangent of 1 divided by n divided by 1 divided by n is equal to 1, and we need to note that this is an element of parentheses of 0, infinity. And we need to note that the series. Evaluated from n is equal to 1. To infinity of tangent of 1 divided by n and the sum evaluated from n is equal to 1 to infinity of 1 divided by n have the same convergence behavior, so we need to note that based on this information the harmonic series. The sum evaluated from n is equal to 1 to infinity of 1 divided by n. Diverges and because this diverges, therefore that will mean that the sum evaluated from n equals 1 to infinity of tangent of 1 divided by n will also diverge. So that means our final answer is diverges. Hooray, we did it. So going back to the top to look at our multiple choice answers, we can safely say that our final answer without a doubt will be multiple choice answer letter B diverges, and that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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 ∞) sin (1/n)

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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 ∞) sin (1/n)