Taylor Series Day 4

Chad Gilliland
24 Feb 201412:52
EducationalLearning
32 Likes 10 Comments

TLDRIn this educational video, the instructor discusses the convergence of infinite series, focusing on three key series: e^x, sin(x), and a geometric series. They explore the properties of these series, demonstrate how to determine their convergence, and apply calculus techniques to find derivatives and integrals of series representations. The session concludes with an example of finding the sum of an infinite series using the McLaurin series.

Takeaways
  • 📚 The script is a lesson on determining the sum of convergent series, specifically focusing on series that are well-known and can be identified by their general term.
  • 🔍 It introduces the concept of the ratio test to prove convergence but emphasizes recognizing patterns in series to identify their sums.
  • 🌟 The first series discussed is e^x, which is a convergent series with the general term x^n/n!, and it equals e^x for all real numbers x.
  • 📉 The second series is the sine series, s(x), which is an alternating series with the general term (2n+1)/ (2n+1)!, and it equals sin(x) for all real numbers.
  • 🔢 The third series is a geometric series with alternating signs, which converges due to its structure and the absolute value of the common ratio being less than one.
  • 📝 A theorem is presented that explains how to take derivatives and integrals of power series without changing the radius of convergence, but possibly altering the interval of convergence.
  • 🔑 The script provides an example of finding the Maclaurin series for f(x) = 1/(1-x) by identifying its first few terms and the general term.
  • ✅ The convergence of the series is discussed, noting that geometric series do not converge at their endpoints, and the interval of convergence is determined.
  • 📈 The derivative of the series is found by differentiating term by term, and the resulting series is used to represent a new function, which is 1/(1-x^2).
  • 🧠 The integral of the series is calculated by integrating term by term, and the resulting series is used to represent the integral of f(t) from 0 to x, which is a sum of powers of t.
  • 📊 The script concludes with the sum of the infinite series being determined by evaluating the integral of 1/(1-x), which is the natural logarithm of the absolute value of (1-x), with the interval of convergence being between -1 and 1.
Q & A

  • What is the main topic of the video script?

    -The main topic of the video script is the practice of determining the convergence of infinite series and identifying what they converge to, with a focus on specific series such as exponential over a factorial, alternating series with factorials, and geometric series.

  • What is the first series mentioned in the script that the instructor believes will converge?

    -The first series mentioned is an infinite sum with a general term of an exponential over a factorial, specifically 3^n / n!, which the instructor believes will converge due to its form.

  • What does the series 3^n / n! converge to?

    -The series 3^n / n! converges to e^3, which is e (Euler's number, approximately 2.71828) raised to the power of 3.

  • What is the second series discussed in the script, and what does it represent?

    -The second series discussed is an alternating series with general term (2 * (2n + 1)) / (2n + 1)!, which is a representation of the sine function, sin(x), where x has been replaced with 2.

  • What is the third series in the script, and what is its convergence behavior?

    -The third series is a convergent alternating geometric series with a general term of (-2/3)^n. It converges because the absolute value of the common ratio (-2/3) is less than 1, and it is an alternating series with terms approaching zero and decreasing in magnitude.

  • What theorem is mentioned in the script regarding the derivative and integral of a power series?

    -The theorem mentioned allows for the term-by-term differentiation and integration of a power series. It states that the radius of convergence remains the same after these operations, although the interval of convergence might change.

  • How does the instructor demonstrate the application of the theorem on the function 1/(1 - x)?

    -The instructor demonstrates by showing the first four terms of the series expansion of 1/(1 - x), then discusses the values of x for which the series converges, and finally applies the theorem to find the Maclaurin series for the derivative of the function.

  • What is the sum of the series found in part B of the script?

    -The sum of the series found in part B, which is the derivative of the function 1/(1 - x), is 9/4 when x is replaced with 1/3.

  • How does the instructor find the sum of an infinite series that doesn't match any known series?

    -The instructor uses the derivative found in part B and rewrites the series in a form that represents a known function. By integrating this function and evaluating it at specific points, the instructor determines the sum of the infinite series.

  • What is the final sum of the infinite series discussed in the script?

    -The final sum of the infinite series, after integrating and evaluating at x = 1/3, is the natural logarithm of 3, or ln(3), because the series converges to -ln(1 - 1/3).

  • What is the significance of checking the endpoints in the interval of convergence for a power series?

    -Checking the endpoints is important to determine if the series converges at the boundary values of the interval, which can affect the domain of the function represented by the series.

Outlines
00:00
📚 Understanding Convergent Series and Their Sums

In this paragraph, the speaker introduces the topic of convergent series and how to determine their sums. They discuss three specific series that are commonly known: the exponential series for e^x, the alternating series for sin(x), and a geometric series. The speaker explains that the exponential series converges for all real numbers and equals e^x, where x is replaced by 3 in the example. The alternating series for sin(x) is also described, emphasizing that it converges for all real numbers. Finally, the geometric series is mentioned, with a focus on its convergence criteria and the formula for its sum. The speaker uses these examples to demonstrate how to recognize and calculate the sums of convergent series.

05:02
🔍 Deriving Series for Derivatives and Integrals

This paragraph delves into the process of deriving series for derivatives and integrals. The speaker starts by discussing the convergence of a geometric series and how to use the McLaurin series to find the derivative of a function. They illustrate this with the function 1/(1-x), showing how to derive its series and determine the range of x for which it converges. The speaker then moves on to integrating series, explaining how to integrate term by term and how the interval of convergence might change. They use the example of the integral from 0 to x of f(t) dt, demonstrating how to integrate each term and adjust for the endpoints. The speaker emphasizes the importance of checking the endpoints and provides an example of how an alternating series can converge at an endpoint.

10:02
🧩 Summing Infinite Series and Applying Theorems

In the final paragraph, the speaker focuses on summing infinite series and applying theorems related to power series. They discuss a theorem that allows for the differentiation and integration of power series without changing the radius of convergence. The speaker uses the example of the function 1/(1-x) to demonstrate how to find the sum of its derivative series and integral series. They show how to rewrite the series in a more recognizable form and calculate the sum by substituting specific values. The speaker also highlights the importance of checking the endpoints for convergence and provides examples of how the interval of convergence can change after integration. The paragraph concludes with a teaser for the next class, where these concepts will be further practiced.

Mindmap
Keywords
💡Convergent Series
A convergent series is a sequence of numbers where the sum of all terms approaches a finite limit as more terms are added. In the script, the speaker discusses methods to determine if a series converges and how to find its sum. For example, the series involving \(3^n / n!\) converges to \(e^3\).
💡Exponential
Exponential refers to a mathematical function involving a constant raised to the power of a variable. In the video, the speaker identifies a series with an exponential term over a factorial, recognizing it as the series for \(e^x\). Specifically, the term \(3^n / n!\) is identified as part of the series for \(e^3\).
💡Factorial
A factorial, denoted by \(n!\), is the product of all positive integers up to \(n\). It is used frequently in series expansions. In the script, factorials appear in the denominators of the terms of the series being discussed, such as \(3^n / n!\) and the alternating series \((-1)^n \cdot x^{2n+1} / (2n+1)!\).
💡Ratio Test
The ratio test is a method used to determine the convergence of a series by examining the ratio of successive terms. While the speaker mentions that they could use the ratio test to prove convergence, they proceed with known results instead. The test involves comparing the limit of the absolute value of the ratio of consecutive terms to 1.
💡Series for \(e^x\)
The series for \(e^x\) is a well-known infinite series represented by \( \sum_{n=0}^{\infty} \frac{x^n}{n!} \). In the script, the speaker uses this series to identify that \(3^n / n!\) converges to \(e^3\), illustrating how recognizing standard series can help determine the sum of more complex series.
💡Alternating Series
An alternating series is a series where the terms alternate in sign. The script discusses the series \((-1)^n \cdot x^{2n+1} / (2n+1)!\) which represents the sine function, \( \sin(x) \). Alternating series are known for their convergence properties, especially when terms decrease in absolute value.
💡Geometric Series
A geometric series is a series with a constant ratio between successive terms. The script includes an example where the series involves a ratio of \( -2/3 \), leading to a sum found using the formula \( \frac{a}{1-r} \). Geometric series are fundamental in determining convergence based on the common ratio.
💡Power Series
A power series is an infinite series in the form \( \sum_{n=0}^{\infty} a_n (x-c)^n \), where \( a_n \) are coefficients and \( c \) is the center of the series. The script discusses properties of power series, such as their radius and interval of convergence, and how differentiation and integration affect them.
💡Radius of Convergence
The radius of convergence is the range of values for which a power series converges. In the video, the speaker explains that the radius of convergence remains unchanged when differentiating or integrating a series, though the interval of convergence might change by including or excluding endpoints.
💡Maclaurin Series
The Maclaurin series is a special case of the Taylor series centered at zero. The script features an exercise where the speaker finds the Maclaurin series for the derivative of a function, using the series expansion for \( \frac{1}{1-x} \) and its derivatives to illustrate the process.
💡Geometric Series Sum Formula
The geometric series sum formula \( \frac{a}{1-r} \) is used to find the sum of an infinite geometric series with the first term \( a \) and common ratio \( r \), provided \( |r| < 1 \). The script applies this formula to a series with a ratio of \( 2/3 \), demonstrating the convergence and calculation of the sum.
💡Harmonic Series
The harmonic series is an infinite series \( \sum_{n=1}^{\infty} \frac{1}{n} \) which diverges. In the video, the speaker references the harmonic series when checking the endpoints for convergence, illustrating its divergence despite terms approaching zero.
Highlights

Introduction to the practice of determining the sum of a convergent series.

Explanation of a convergent series with an exponential over a factorial.

Recognition of a series for e to the X and its convergence properties.

Description of the series for e to the X and its equivalence to e raised to the power of X.

Identification of a series that converges to e cubed.

Introduction to an alternating series with factorials and its relation to the sine function.

Explanation of the series for S of X and its convergence to sin of X.

Identification of a series that converges to sin of two.

Discussion on the convergence of an alternating geometric series and its properties.

Explanation of a geometric series and its convergence criteria.

Description of a theorem related to power series, derivatives, and integrals.

Application of the theorem to find the Maclaurin series for the derivative of a function.

Calculation of the sum of an infinite series using the derivative of a known function.

Integration of a series and the determination of its convergence interval.

Explanation of how integrating a series can affect its convergence interval.

Conclusion on the sum of an infinite series related to the natural logarithm function.

Transcripts

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Related Tags
Series ConvergenceCalculus PracticeExponential SeriesFactorial SeriesGeometric SeriesAlternating SeriesRatio TestDerivative SeriesIntegral SeriesConvergence Theorem