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

in hive-181335 •  12 days ago 

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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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Infinite Sequences: Limits, Squeeze Theorem, Fibonacci Sequence & Golden Ratio + MORE:

Infinite Series: Definition, Examples, Geometric Series, Harmonics Series, Telescoping Sum + MORE:

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In this video I show how we can use the integral of the function corresponding to the series to determine if the series converges or not without having to deal with the series directly. Similarly, an estimate of the sum itself can be made by considering the remainder in terms of the integral as well.

View video notes on the Hive blockchain: