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!
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Infinite Sequences: Limits, Squeeze Theorem, Fibonacci Sequence & Golden Ratio + MORE: https://peakd.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more
Infinite Series: Definition, Examples, Geometric Series, Harmonics Series, Telescoping Sum + MORE: https://peakd.com/mathematics/@mes/infinite-series-definition-examples-geometric-series-harmonics-series-telescoping-sum-more
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