Use the divergence test to determine whether the infinite series
As always, we apply the divergence theorem by evaluating a limit as tends to infinity. In this case we find
Therefore, because does not tend to zero as k tends to infinity, the divergence test tells us that the infinite series diverges.
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Remembering What arctan Looks Like
The arctan function is the inverse of the tan function. One way of remembering what it looks like is to remember that the graph of the inverse of a function can be obtained by reflecting it through the straight line y = x. The two functions are shown in the figure below.