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JH
Numerade Educator

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Problem 39 Medium Difficulty

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \arctan n $

Answer

$$\frac{\pi}{2} \neq 0$$

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Video Transcript

that's determine whether the Siri's convergence or diversions and then if it convergence will go ahead and find the sum, too. So some actual there? Possibly. So. Let's look at the limit and goes to infinity of armed men. The end. So before I write this answer, let me just refresh your memory. So to define Arc Tan, we just take the usual tan function. Believers just restricted between negative pyros to and power for two. So here's our usual tangent graph. So this graph, this will have an inverse because it passes the horizontal light test. And so, to graph the inverse recall that you just swap the X and Y values so these Assam talks will become horizontal ass from tops. So there's negative high over too. And then up here we have pie over, too. And now that we go ahead and let's guidance, for example, this point zero zero, if you switch those coordinates, you still get zero zero, so that will still be here. So go ahead and take each point on here and switched the X No, why in the following graph will be the this green one right here. So, by looking at the graph of arc tend, let's say ends. Of course, this girl, I'm drawing the continuous version. So let me just put an X here. We could see that the graft gets closer and closer to a pie or two. So that answers this question. Up here, our skin gets close to the survivors. Who and let's note that this is not equal to zero. And then we're basically finished here because we have what your book calls the diversions test. And this is if the limit as N goes to infinity of A M does not exist. No or if it does exist. But it's not equal to zero. That's what's happening in our case that exists. But it's not zero. Then we have that the Siri's diverges. Therefore, in our case, since we have that the limit is on zero, we have in conclusion and equals one to infinity. Our ten end is a diversion by the diversions test, and that's your final answer