11–86. Applying convergence tests Determine whether the follow...

Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞)tan⁻¹(1 / √k)

Accepted Answer

Hello there. Today we're gonna 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. Consider the series, the sum evaluated from J equals 1 to positive infinity of inverse tangent of 2 divided by J. Does this series converge? Awesome. So it appears for this particular prompt we're asked to take the series that is provided to us by the prom itself, and we're asked to determine does this series converge. So does it a, yes, the series converges, or B, no, the series diverges. So now that we know what we're ultimately trying to solve for, Our first step that we need to take is we need to be able to look at our series, and we need to be able to recall and recognize that this is a positive term series and because it's a positive term series, we can therefore recall and use the limit comparison test. And we can determine that this is an appropriate method to use to solve this particular problem. So as we should recall, for a limit comparison test for positive sequences, a subscript M and B subscript M, we can say that if for these positive sequences lowercase l is equal to the limit, as m approaches positive infinity of a subscript M divided by B subscript M. And this exists, and 0 is less than L and L is less than positive infinity, then that will mean that the sum of a subscript M and the sum of B subscript M either both converge or diverge due to this limit comparison test. So, with that in mind, our first step to solve this problem, or rather the first major step, is we need to choose a comparison series. So with that in mind, for large values of J, that would mean that Inverse tangent, so we're gonna take this inverse tangent of X will be roughly X. So let us compare with B subscript J is equal to 1 divided by J. So that will mean our second step now is we need to compute the limit. So we need to take the limit as J approaches positive infinity of a divided by J. Or rather, sorry, A subscript J divided by B subscript J is equal to the limit. So once again, this will be equal to the limit, once again, as J approaches positive infinity of inverse tangent of 2 divided by J. All divided by 1 divided by J, which will be equal to the limit as J approaches positive infinity of J multiplied by inverse tangent of 2 divided by J. So now we need to set X. is equal to 1 divided by J as it approaches 0. So then we can go ahead and write that. Once again, so we're noting that we are setting X is equal to 1 divided by J as it approaches 0, then we can go ahead and write that J multiplied by inverse tangent of 2 divided by J is equal to inverse tangent of 2 X divided by X as it approaches 2, as X approaches 0. Fantastic. So that will mean for our 3rd and final step we can now officially apply the limit comparison test and the limit, as we should recall, so our limit is equal to 2 and 2 is greater than 0, so that makes it finite. And since the sum of B subscript J is equal to the sum of 1 divided by J. Diverges, which I'll say D for short for diverges, because once again that this is a harmonic series, which I'll call HS for short. So once again, we know that the limit is 2 and 2 is greater than 0, so this makes it finite. So let's make a note of that, since it's very important. It's finite and because it's finite, The sum of B subscript J will be equal to the sum of 1 divided by J, which will mean that this is diverging because it's a harmonic series. So that means that the limit comparison test implies that the sum of a subscript J will also diverge as a result. So now that we know that the series is diverging, our final answer without a doubt will have to be multiple choice answer letter B, which once again is no, the series diverges, and that is our correct and final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)tan⁻¹(1 / √k)

Key Concepts

Convergence of Infinite Series

Comparison Test and Limit Comparison Test

Behavior of arctan(x) for Small Arguments

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