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Week 6: Taylor Series
(collapsed, click to expand)- Completed 6.00 What is this last week about? [2:10]
- Completed 6.01 What is better than a linear approximation? [14:12]
- Completed 6.02 What is the Taylor series for f around zero? [10:12]
- Completed
6.03 What is the Taylor series for f centered around a? [4:39]
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- Completed
6.04 What is the Taylor series for sin around zero? [4:36]
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- Completed 6.05 What is Taylor's theorem? [11:42]
- Completed
6.06 Why is the radius of convergence of 1/(1+x^2) so small?
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- Completed
6.07 How is Taylor's theorem like a souped up version of the mean value theorem? [8:04]
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- Completed
6.08 Approximately, what is cos x when x is near zero? [8:14]
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- Completed
6.09 How do Taylor series provide intuition for limits? [11:03]
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- Completed 6.10 What is a real analytic function? [4:54]
- Completed
6.11 How are real analytic functions sometimes like holograms? [4:24]
Forum for 6.11 How are real analytic functions sometimes like holograms? [4:24]Subtitles (text) for 6.11 How are real analytic functions sometimes like holograms? [4:24]Subtitles (srt) for 6.11 How are real analytic functions sometimes like holograms? [4:24]Video (MP4) for 6.11 How are real analytic functions sometimes like holograms? [4:24]
Week 5: Power series
(collapsed, click to expand)- Completed 5.00 What are power series? [3:03]
- Completed
5.01 For which values does a power series converge? [3:47]
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- Completed 5.02 Why does a power series converge absolutely? [4:53]
- Completed
5.03 How complicated might the interval of convergence be? [5:15]
Forum for 5.03 How complicated might the interval of convergence be? [5:15]Subtitles (text) for 5.03 How complicated might the interval of convergence be? [5:15]Subtitles (srt) for 5.03 How complicated might the interval of convergence be? [5:15]Video (MP4) for 5.03 How complicated might the interval of convergence be? [5:15]
- Completed 5.04 How do I find the radius of convergence? [3:26]
- Completed
5.05 What if the radius of convergence is infinite? [4:02]
Forum for 5.05 What if the radius of convergence is infinite? [4:02]Subtitles (text) for 5.05 What if the radius of convergence is infinite? [4:02]Subtitles (srt) for 5.05 What if the radius of convergence is infinite? [4:02]Video (MP4) for 5.05 What if the radius of convergence is infinite? [4:02]
- Completed 5.06 What if the radius of convergence is zero? [3:26]
- Completed 5.07 What is a power series centered around a? [4:06]
- Completed 5.08 Can I differentiate a power series? [3:30]
- Completed 5.09 Can I integrate a power series? [6:18]
- Completed
5.10 Why might I believe I have a power series for e^x? [4:13]
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- Completed 5.11 What happens if I multiply two power series? [9:27]
- Completed 5.12 What happens if I transform 1/(1-x)? [6:59]
- Completed 5.13 What is a formula for the Fibonacci numbers?
Week 4: Alternating series
(collapsed, click to expand)- Completed 4.00 What is Week 4 all about? [1:14]
- Completed
4.01 Why have we been assuming the terms are positive? [3:35]
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- Completed
4.02 Why do absolutely convergent series just plain converge? [5:20]
Forum for 4.02 Why do absolutely convergent series just plain converge? [5:20]Subtitles (text) for 4.02 Why do absolutely convergent series just plain converge? [5:20]Subtitles (srt) for 4.02 Why do absolutely convergent series just plain converge? [5:20]Video (MP4) for 4.02 Why do absolutely convergent series just plain converge? [5:20]
- Completed
4.03 Why is absolute convergence an important concept? [3:37]
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- Completed 4.04 What is conditional convergence? [2:51]
- Completed 4.05 What is an alternating series? [4:09]
- Completed 4.06 What is the alternating series test? [11:03]
- Completed
4.07 How should I go about checking the convergence of a series? [1:42]
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- Completed
4.08 Why is monotonicity important in the alternating series test? [5:43]
Forum for 4.08 Why is monotonicity important in the alternating series test? [5:43]Subtitles (text) for 4.08 Why is monotonicity important in the alternating series test? [5:43]Subtitles (srt) for 4.08 Why is monotonicity important in the alternating series test? [5:43]Video (MP4) for 4.08 Why is monotonicity important in the alternating series test? [5:43]
- Completed 4.09 Why are alternating series important? [6:09]
- Completed 4.10 Why is e irrational? [8:41]
- Completed 4.11 When do two series share the same fate? [5:53]
- Completed
4.12 Why can people get away with writing sum_n a_n? [4:31]
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- Completed 4.13 Why is this all so vague... or coarse? [6:05]
- Completed
4.14 What happens if I rearrange the terms in a conditionally convergent series? [7:29]
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Week 3: Convergence tests
(collapsed, click to expand)- Completed 3.00 What will happen in Week 3? [1:22]
- Completed 3.01 Does sum n^5 / 4^n converge? [13:38]
- Completed 3.02 What does the ratio test say? [7:51]
- Completed 3.03 Does the ratio test always work? [5:34]
- Completed 3.04 Does sum n! / n^n converge? [9:05]
- Completed 3.05 How does n! compare to n^n? [6:12]
- Completed 3.06 Why don't I love the root test? [3:40]
- Completed
3.07 How can integrating help us to address convergence? [12:26]
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- Completed
3.08 How else can I show the harmonic series diverges? [5:05]
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- Completed 3.09 Does sum 1/n^p converge? [8:06]
- Completed 3.10 Does sum 1/(n log n) converge? [7:39]
- Completed
3.11 How far out can you build a one sided bridge? [10:44]
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Week 2: Series
(collapsed, click to expand)- Completed 2.00 What happens in Week 2? [1:20]
- Completed 2.01 What does ∑ aₙ = L mean? [3:53]
- Completed 2.02 Why does ∑ₖ₌₀∞ 1/2ᵏ = 2? [5:20]
- Completed 2.03 What is a geometric series? [9:30]
- Completed 2.04 What is the value of ∑ₖ₌ₙ∞ rᵏ? [7:05]
- Completed 2.05 What is the sum of a telescoping series? [10:10]
- Completed
2.06 Does the series ∑ n/(n+1) converge or diverge? [10:14]
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- Completed
2.07 Does the series 1 + 1/2 + 1/3 + ⋯ converge or diverge? [6:39]
Forum for 2.07 Does the series 1 + 1/2 + 1/3 + ⋯ converge or diverge? [6:39]Subtitles (text) for 2.07 Does the series 1 + 1/2 + 1/3 + ⋯ converge or diverge? [6:39]Subtitles (srt) for 2.07 Does the series 1 + 1/2 + 1/3 + ⋯ converge or diverge? [6:39]Video (MP4) for 2.07 Does the series 1 + 1/2 + 1/3 + ⋯ converge or diverge? [6:39]
- Completed 2.08 Does ∑ sin² k / 2ᵏ converge or diverge? [3:51]
- Completed 2.09 What is the comparison test? [8:26]
- Completed
2.10 How can grouping make the comparison test even better? [6:19]
Forum for 2.10 How can grouping make the comparison test even better? [6:19]Subtitles (text) for 2.10 How can grouping make the comparison test even better? [6:19]Subtitles (srt) for 2.10 How can grouping make the comparison test even better? [6:19]Video (MP4) for 2.10 How can grouping make the comparison test even better? [6:19]
- Completed 2.11 What is ∑ 1/n² ? [9:00]
- Completed 2.12 In what sense does 0.99999⋯ equal 1? [5:05]
- Completed 2.13 In what sense is ∑ 9⋅10ⁿ meaningful? [9:40]
Week 1: Sequences
(collapsed, click to expand)- Completed 1.00 How can I succeed in this course? [1:07]
- Completed 1.01 What is a sequence? [4:05]
- Completed 1.02 How is a sequence presented? [5:45]
- Completed
1.03 Can the same sequence be presented in different ways? [3:39]
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- Completed
1.04 How can we build new sequences from old sequences? [3:24]
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- Completed 1.05 What is an arithmetic progression? [2:18]
- Completed 1.06 What is a geometric progression? [4:26]
- Completed 1.07 What is the limit of a sequence? [4:42]
- Completed 1.08 Visually, what is the limit of a sequence? [1:43]
- Completed 1.09 Is it easy to find the limit of a sequence? [8:03]
- Completed 1.10 For some epsilon, how large need N be? [4:09]
- Completed
1.11 How do sequences help with the square root of two? [3:02]
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- Completed 1.12 When is a sequence bounded? [8:27]
- Completed 1.13 When is a sequence increasing? [4:47]
- Completed 1.14 What is the Monotone Convergence Theorem? [1:47]
- Completed
1.15 How can the Monotone Convergence Theorem help? [5:44]
Forum for 1.15 How can the Monotone Convergence Theorem help? [5:44]Subtitles (text) for 1.15 How can the Monotone Convergence Theorem help? [5:44]Subtitles (srt) for 1.15 How can the Monotone Convergence Theorem help? [5:44]Video (MP4) for 1.15 How can the Monotone Convergence Theorem help? [5:44]
- Completed
1.16 Is there a sequence that includes every integer? [5:57]
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1.17 Is there a sequence that includes every real number? [5:25]
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