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Infinite series 9 Cauchy Product of Infinite Series Basic Real Analysis
>> YOUR LINK HERE: ___ http://youtube.com/watch?v=04n7FyYcHkQ
We look at three examples: (i) Multiplication of polynomials, (ii) the exponential series and the property that exp(x+y)=exp(x).exp(y) and (iii) the term-wise differentiation of the geometric (power) series sum(x^n). These motivate the definition of the Cauchy product of two infinite series. The main result is that the Cauchy product of two absolutely convergent series is absolutely convergent and that the sum of the Cauchy product is the product of the sums of the two series. We then apply to derive the results of Examples (ii) and (iii). • This result is very useful and suffices for most of situations that arise in analysis. There is a more general result known as Merten's theorem. Its proof is a good training for any aspiring analyst. Watch • Infinite Series 10: Cauchy Product an... • This video is linked towards the end of the video. • The proof of the main result, though easy, is an example of an argument with finesse. Hope you enjoy it! • After recording the video, I have typed the entire session. Send a request to [email protected] if you wish to receive a pdf version of this article.
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