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Title: Sum Formulas for Sine and Cosine
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Series: Fourier Transform
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Chapter: Fourier Series
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YouTube-Title: Fourier Transform 11 | Sum Formulas for Sine and Cosine
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Bright video: Watch on YouTube
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Dark video: 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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Python file: Download Python file
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: ft11_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 Introduction
01:00 Statement for Cosine Formula
01:51 Note about finite sum of Cosine functions
07:10 Lemma about Sine Formula
08:40 Proof of Lemma
17:13 Visualization of Lemma
18:00 Theorem (Cosine Formula)
18:30 Proof of Theorem
22:22 Applying Weierstrass M-Test
23:57 Find integration constant
25:23 Credits
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $x \in (0,2 \pi)$. What is correct for $\sum_{k=1}^\infty \frac{\sin(kx)}{k}$?
A1: $\frac{\pi - x }{2}$
A2: $\frac{\pi + x }{2}$
A3: $\frac{x }{2}$
A4: $\frac{\pi }{2}$
Q2: Let $x \in [0,2 \pi]$. What is correct for $\sum_{k=1}^\infty \frac{\cos(kx)}{k^2}$?
A1: $\frac{(x-\pi)^2}{4} - \frac{\pi^2}{12}$
A2: $\frac{x^2}{4} - \frac{\pi^2}{12}$
A3: $\frac{(x-\pi)^3}{4} - \frac{\pi^2}{6}$
A4: $\frac{x-\pi}{2} - \frac{\pi^2}{2}$
Q3: Use the cosine formula from before to calculate $\sum_{k=1}^\infty \frac{1}{k^2}$.
A1: $\frac{\pi^2}{6}$
A2: $ -\frac{\pi^2}{6}$
A3: $- \frac{\pi^2}{4}$
A4: $ \frac{\pi^2}{4}$
A5: $ \frac{\pi}{8}$
A6: $ - \frac{\pi}{8}$
A7: $- \frac{\pi^2}{12}$
A8: $ \frac{\pi^2}{12}$
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Last update: 2025-09
