• Title: Cauchy Product

  • Series: Real Analysis

  • Chapter: Infinite Series

  • YouTube-Title: Real Analysis 22 | Cauchy Product

  • Bright video: https://youtu.be/tRa0Ex_0Yfo

  • Dark video: https://youtu.be/upQzHlMCus0

  • Quiz: Test your knowledge

  • PDF: Download PDF version of the bright video

  • Dark-PDF: Download PDF version of the dark video

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  • Subtitle on GitHub: ra22_sub_eng.srt missing

  • Timestamps

    00:00 Intro

    00:38 Looking at finite sums

    02:30 Definition Cauchy Product

    04:07 Theorem about abs. convergence

    04:49 Example for the theorem

    07:51 Credits

  • Subtitle in English (n/a)
  • Quiz Content

    Q1: What is the correct definition for the Cauchy product of two series $\displaystyle \sum_{k = 0}^\infty a_k$ and $\displaystyle \sum_{k = 0}^\infty b_k$?

    A1: It’s the series $\displaystyle \sum_{k = 0}^\infty c_k$ with $c_k = \displaystyle \sum_{\ell = 0}^k a_{\ell} b_{\ell}$.

    A2: It’s the series $\displaystyle \sum_{k = 0}^\infty c_k$ with $c_k = \displaystyle \sum_{\ell = 0}^k a_{\ell} b_{\ell-k}$.

    A3: It’s the series $\displaystyle \sum_{k = 0}^\infty c_k$ with $c_k = \displaystyle \sum_{\ell = 0}^k a_{\ell} b_{k-\ell}$.

    A4: It’s the series $\displaystyle \sum_{k = 0}^\infty c_k$ with $c_k = \displaystyle \sum_{\ell = 0}^k a_{k} b_{\ell-k}$.

    A5: It’s the series $\displaystyle \sum_{k = 0}^\infty c_k$ with $c_k = \displaystyle \sum_{\ell = 0}^k a_{\ell-k} b_{k-\ell}$.

    Q2: Is the Cauchy product for two convergent series $\displaystyle \sum_{k = 0}^\infty a_k$ and $\displaystyle \sum_{k = 0}^\infty b_k$ also convergent?

    A1: No, never!

    A2: Yes, always!

    A3: No, not in general!

    Q3: What is the correct definition of the exponential function $\exp$.

    A1: $\exp(x) := \displaystyle \sum_{k = 1}^\infty \frac{x^k}{k!}$

    A2: $\exp(x) := \displaystyle \sum_{k = 0}^\infty \frac{x^k}{k!}$

    A3: $\exp(x) := \displaystyle \sum_{k = 0}^\infty \frac{x^k}{k}$

    Q4: What is a correct property of the exponential function $\exp$?

    A1: $\exp(x+y) = \exp(x)\cdot \exp(y)$

    A2: $\exp(x \cdot y) = \exp(x) \cdot \exp(y)$

    A3: $\exp(x+y) = \exp(x)+\exp(y)$

    A4: $\exp(x \cdot y) = \exp(x)+\exp(y)$

    A5: $\exp(x+y) = \exp(x) - \exp(y)$

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