Infinite Sequences and Series: The Integral Test and Estimate of Sums

In this video, I go over further into Infinite Sequences and Series and this time look at The Integral Test and Estimating Sums. Often times it is not easy or even possible to formulate a simple function for the n-th partial sum, thus making it very difficult to directly compute the total sum of the series. Thus, other methods are required to first determine if a given series will converge or diverge but without having to do deal directly with the series. I go over one such method, The Integral Test, which geometrically shows that if a series can be written as a function of x and it is decreasing and continuous as x gets large, then the divergence or convergence of the integral of the function itself (i.e. the area under the curve) correlates to the divergence or convergence of the series. A similar approach is shown for estimating the value of a convergent sum and involves considering the minimum and maximum bounds of the remainder in terms of the area or integral of the function as well. This is another in-depth dive into infinite sequences and series so hope you enjoy!

1. @ 1:30 - Introduction to The Integral Test and Estimate of Sums
2. @ 22:20 - The Integral Test
   • @ 27:39 - Example 1
   • @ 37-20 - Example 2: P-Series
     • @44:06 - Theorem 1: P-Series
   • @44:35 - Example 3
   • @ 46:56 - Example 4
3. @ 1:00:27 - Estimating the Sum of a Series
   • @ 1:06:46 - Theorem 2: Remainder Estimate for the Integral Test
   • @1:08:58 - Example 5
   • @1:20:12 - Theorem 3
   • @ 1:21:59 - Example 6
4. @ 1:29:31 - Proof of the Integral Test

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