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Title: Cauchy Product
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Series: Real Analysis
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Chapter: Infinite Series
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YouTube-Title: Real Analysis 22 | Cauchy Product
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Bright video: Watch on YouTube
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Dark video: Watch on YouTube
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Ad-free video: Watch Vimeo video
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Original video for YT-Members (bright): Watch on YouTube
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Original video for YT-Members (dark): Watch on YouTube
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Exercise Download PDF sheets
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: ra22_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Download dark video: Link on Vimeo
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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
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Subtitle in English (n/a)
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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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Last update: 2025-01
