The Integral Test + Example on Convergence of a Series

In this video, I go over the Integral Test, which states that a series converges only if its corresponding improper integral also converges. Likewise, if the improper integral diverges, then the series also diverges. This stems from my earlier introduction video illustrating how the sum of an infinite series is directly tied to the area under the curve of the equation for the terms of the series. I illustrate this with an example on the series with terms 1/(n²+1), which converges since its improper integral converges and equals π/4.

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#math #calculus #series #integral #trigonometry

Timestamps

• The Integral Test – 0:00
  • It is not necessary to start the integral or series at n = 1 – 2:05
  • It is not necessary for the function to be always decreasing, only that it is ultimately decreasing – 2:54
• Example 1: Test convergence of series with terms 1/(n²+1) – 5:09
  • Recall derivative of inverse tangent – 6:28
  • Recall exact trigonometric ratios for tangent – 8:29
  • Recall the graph for tangent and inverse tangent – 10:08
  • Improper integral is convergent, thus the series is convergent by the Integral Test – 14:12

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