Introduction to the Integral Test and Estimating Sums of Infinite S...

In this video, I go over an introduction into estimating sums of series by examining whether their corresponding improper integrals converge or diverge, thus providing the basis for the Integral Test. Often times, there is no exact formula for the sum of a series, so we need to obtain ways of estimating their sums. I first illustrate this with the series of reciprocals of squares of positive integers (terms 1/n^2) and then look at the same but with square roots (terms 1/n^(1/2)). In both cases, I calculate the partial sums directly via Microsoft Excel, and then illustrate them geometrically as sums of rectangles contacting their corresponding curves. The first series converges since it is less than the area under the curve of a convergent improper integral. The second series diverges since it is greater than the area under a divergent improper integral.

In the next video, I will use these insights to discuss the Integral Test for convergence or divergence of infinite series.

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Timestamps

Finding the exact sum of a series is often difficult, so we need to estimate it – 0:00
Analyzing the squares of reciprocals of positive integers – 1:33
Analyzing the sum of the series with Microsoft Excel since there is no exact formula for the sum – 2:40
Geometric argument by finding the area under the curve – 3:40
The sum of the rectangles is equal to the sum of the series – 6:18
Recall that this improper integral is convergent and has a value of 1 – 7:45
Partial sums are less than the area under the curve, which is 2 – 9:05
The series is bounded, increasing, and therefore convergent by the monotonic sequence theorem – 10:30
The exact sum of the series is equal to pi^2/6 = 1.644934... – 13:33
The series of square-roots of reciprocals of positive integers appears diverging – 13:57
The series is greater than the area under the curve, whose improper integral is divergent – 18:43
The series is thus divergent as well – 20:32