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Title: Integral Test for Series
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Series: Real Analysis
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Chapter: Riemann Integral
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YouTube-Title: Real Analysis 62 | Integral Test for Series
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Bright video: https://youtu.be/iCEKW61q5hk
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Dark video: https://youtu.be/rbPhKSd5mA8
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Ad-free video: Watch Vimeo video
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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: ra62_sub_eng.srt missing
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $f: [0,\infty) \rightarrow [0,\infty)$ be monotonically decreasing. The integral $\int_0^{\infty} f(x) dx$ converges if and only if $\sum_{k=0}^\infty f(k)$ converges.
A1: Correct!
A2: Not correct!
A3: Only correct if $f$ is never zero.
Q2: Let $f: [0,\infty) \rightarrow [0,\infty)$ be monotonically decreasing where the integral $\int_0^{\infty} f(x) dx$ converges. Which inequality is always correct?
A1: $$\sum_{k=0}^\infty f(k) - \int_0^{\infty} f(x) dx \geq f(0)$$
A2: $$\sum_{k=0}^\infty f(k) - \int_0^{\infty} f(x) dx \leq f(0)$$
A3: $$\sum_{k=0}^\infty f(k) - \int_0^{\infty} f(x) dx \geq f(1)$$
Q3: Does the series $\sum_0^{\infty} \frac{1}{\sqrt{k}}$ converge?
A1: Yes!
A2: No!
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Last update: 2025-01
