Infinite Sequences and Series: Representations of Functions as Power Series
In this video I go over further into infinite sequences and series; and in particular expand upon my earlier video on power series. This time I show how we can represent many functions as power series as well as go over a theorem that allows for differentiation and integration of power series term by term. This theorem, as well as this overall concept of representing functions as power series is one of the most important concepts in all of science because it allows for much easier computation of many different functions. In fact many computers and calculators use this technique behind the scenes since they are often much easier to deal with than trying to find exact integrals or derivatives which many times are not simple.
In this video, and at the second half of this video, I also go over an extensive 2 hour long proof of the theorem for term-by-term differentiation and integration of power series. Although this proof is extensive, after searching all over online for the proof, I believe my proof may in fact be the most clear and understandable. Let me know if it helps!
The topics covered in this video are listed below with their time stamps.
- @ 0:00 - Intro
1. @ 2:24 - Representation of Functions as Power Series
- @ 3:17 - Equation 1
- @ 10:48 - Example 1
- @ 26:35 - Example 2
- @ 30:16 - Example 3
2. - @ 37:06 - Differentiation and Integration of Power Series
- @ 38:09 - Theorem 1: Term by Term Differentiation and Integration
- @ 50:50 - Example 4
- @ 54:18 - Example 5
- @ 58:36 - Example 6
- @ 1:11:02 - Example 7: Gregory's Series
- @ 1:25:59 - Leibniz Formula for π
- @ 1:30:30 - Example 8
3. Exercises
- @ 2:11:57 - Exercise 1
- @ 1:55:34 - Exercise 2
- @ 2:26:11 - Exercise 3: Proof of Theorem 1
- @ 2:32:29 - Lemma
- @ 2:34:58 - Lemma Derivative Proof
- @ 3:01:16 - Lemma Integral Proof
- @ 3:08:59 - Theorem 1 Using z = (x - a)
- @ 3:12:41 - Theorem 1a Derivative Proof
- @ 3:24:59 - Brief Overview of the Binomial Theorem
- @ 3:57:33 - Theorem 1b Integral Proof
Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIh_17muwohTwCBpQ45Q?e=pldHwj
View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-representations-of-functions-as-power-series
Related Videos:
Sequences and Series Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz
Infinite Sequences: Limits, Squeeze Theorem, Fibonacci Sequence & Golden Ratio + MORE: https://peakd.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more
Infinite Series: Definition, Examples, Geometric Series, Harmonics Series, Telescoping Sum + MORE: https://peakd.com/mathematics/@mes/infinite-series-definition-examples-geometric-series-harmonics-series-telescoping-sum-more
Infinite Sequences and Series: The Integral Test and Estimate of Sums: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-the-integral-test-and-estimate-of-sums
Infinite Sequences and Series: The Comparison Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-the-comparison-tests
Infinite Sequences and Series: Alternating Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-alternating-tests
Infinite Sequences and Series: Absolute Convergence and the Ratio/Root Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-absolute-convergence-and-the-ratio-root-tests
Infinite Sequences and Series: Strategy for Testing Series: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-strategy-for-testing-series
Infinite Sequences and Series: Power Series: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-power-series .
------------------------------------------------------
SUBSCRIBE via EMAIL: https://mes.fm/subscribe
DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate
Like, Subscribe, Favorite, and Comment Below!
Follow us on:
MES Truth: https://mes.fm/truth
Official Website: https://MES.fm
Hive: https://peakd.com/@mes
MORE Links: https://linktr.ee/matheasy
Email me: contact@mes.fm
Free Calculators: https://mes.fm/calculators
BMI Calculator: https://bmicalculator.mes.fm
Grade Calculator: https://gradecalculator.mes.fm
Mortgage Calculator: https://mortgagecalculator.mes.fm
Percentage Calculator: https://percentagecalculator.mes.fm
Free Online Tools: https://mes.fm/tools
iPhone and Android Apps: https://mes.fm/mobile-apps
-
2:06:53
Math Easy Solutions
1 year ago $0.01 earnedInfinite Sequences and Series: Power Series
1051 -
5:03:52
Math Easy Solutions
1 year ago $0.01 earnedInfinite Sequences and Series: Taylor and Maclaurin Series
711 -
1:02:52
DrOfEng
7 months agoInfinite Sequences and Series, Fast Revision, Worked Examples - Unit 10 - AP Calculus BC
25 -
7:50
Calculus for Engineering
3 years agoSequence & Series (Calculus)
851 -
24:02
ericntunctu
2 years agoPower series and special functions
2 -
25:12
ericntunctu
2 years agoPower series definition for exponential function, logarithmic function, trigonometric function
4 -
33:39
Mathematics Patterns, Sequences and Series
9 months agoGeometric and Arithmetic Series, Sum to infinity Q4 Grade 12 Mathematics Revision
957 -
1:23:46
DrOfEng
8 months agoIntegration and Accumulation of Change, Fast Revision, Worked Examples - Unit 6 - AP Calculus BC
1261 -
18:41
The Solution Manual
2 months agoExponent Problems Made Easy (timestamps in description) #algebra #precalculus #exponents
14 -
30:49
Learn Math with Caveman Chang
2 years agoArithmetic SEQUENCES & Arithmetic SERIES - Introduction
64