Interval of convergence for derivative and integral | Series | AP Calculus BC | Khan Academy

Khan Academy
26 Dec 201608:34
EducationalLearning
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TLDRThis video script delves into the operations of differentiation and integration of power series, illustrating how these can be performed term by term. It explains the concept of the interval of convergence, using the ratio test to determine it for both the original power series and its derivatives and integrals. The script highlights the subtle differences in convergence at the endpoints and encourages viewers to apply these concepts to find the intervals for themselves, noting the harmonic and alternating series as special cases.

Takeaways
  • 📚 Derivatives and integrals of power series can be computed term by term.
  • 🧮 The derivative of a power series f(x) is the sum of the derivatives of each term.
  • ⚖️ Taking the derivative term by term simplifies to the original series shifted by one exponent.
  • ✍️ Integrating a power series term by term involves incrementing the exponent and dividing by the new exponent.
  • 🔄 The ratio test is used to determine the interval of convergence for a power series.
  • 📐 For the given series, the ratio test shows convergence when the absolute value of x is less than 1.
  • ↔️ The interval of convergence for the original series is -1 ≤ x < 1.
  • 🌀 The geometric series converges when the absolute value of the common ratio is less than 1.
  • 🔍 When taking the derivative of the original series, the interval of convergence changes slightly, excluding -1.
  • 🔗 All series derived from the original have the same radius of convergence but may differ at endpoints.
Q & A

  • What does 'term by term' mean when dealing with power series?

    -When taking the derivative or integral of a power series, 'term by term' means performing the operation on each term of the series individually, and then summing the results.

  • How do you find the derivative of a power series?

    -The derivative of a power series is found by differentiating each term of the series with respect to x, resulting in a new series where each term is \( n \times x^{n-1} \), and then summing these differentiated terms.

  • What is the derivative of \( x^n \) with respect to x?

    -The derivative of \( x^n \) with respect to x is \( n \times x^{n-1} \).

  • How does the derivative of a power series affect the terms of the series?

    -The derivative of a power series results in each term's exponent being decreased by one, and the series becomes a sum of these new terms.

  • What is the process for integrating a power series?

    -Integration of a power series is done term by term, incrementing the exponent of each term by one and dividing by the new exponent, and then summing these integrated terms.

  • What is the general form of the integral of a power series term?

    -The integral of a power series term is \( \frac{x^{n+1}}{n+1} \), where n is the original exponent of the term.

  • Why is it important to consider the interval of convergence when working with power series?

    -The interval of convergence is crucial because it defines the range of x values for which the power series converges. Operations like differentiation and integration are only valid within this interval.

  • What is the ratio test and how is it used to determine the interval of convergence?

    -The ratio test is a method used to determine the convergence of a series by examining the limit of the ratio of consecutive terms as n approaches infinity. If the limit is less than 1, the series converges; if greater than 1, it diverges; and if equal to 1, the test is inconclusive.

  • How does the interval of convergence for a power series change when you take its derivative?

    -When taking the derivative of a power series, the interval of convergence may change at the endpoints. The radius of convergence remains the same, but the behavior at the endpoints (whether the series converges or diverges) can differ.

  • What is the interval of convergence for the original power series discussed in the script?

    -The interval of convergence for the original power series is from -1 to 1, not including -1 and 1, because the series diverges at these points.

  • How does the interval of convergence for the derivative of a power series compare to the original series?

    -The interval of convergence for the derivative of a power series is generally the same as the original series, except it does not include the endpoints where the original series diverges, such as -1 and 1 in this case.

  • What is the interval of convergence for the antiderivative or integral of the original power series?

    -The interval of convergence for the antiderivative includes the endpoints -1 and 1, as these points converge due to the nature of the integral operation, even though the original series diverges at these points.

  • How can you determine if a power series converges at the endpoints of its interval of convergence?

    -To determine convergence at the endpoints, one must evaluate the series directly at those points, as the ratio test becomes inconclusive. For example, when x equals 1 or -1, the series must be evaluated to see if it converges or diverges.

Outlines
00:00
📚 Derivatives and Integrals of Power Series

This paragraph discusses the process of taking derivatives and integrals of power series term by term. The instructor explains that the derivative of a power series function f(x), denoted as f'(x), is obtained by differentiating each term of the series, resulting in a new series where each term is x to the power of (n-1). Similarly, integrating a power series involves summing the terms with an incremented exponent, divided by the new exponent, n+1. The instructor emphasizes that these operations are valid within the interval of convergence for the series and invites viewers to determine the interval of convergence for the given series using the ratio test. The ratio test is applied to find the limit of the ratio of consecutive terms, and the series is convergent if the absolute value of x is less than one, divergent if greater than one, and inconclusive if equal to one. Special cases when x equals one or negative one are also discussed, noting that the series diverges at x=1 but converges at x=-1 due to the alternating nature of the terms.

05:02
🔍 Intervals of Convergence for Power Series and Their Derivatives

The second paragraph delves deeper into the intervals of convergence for both the original power series and its derivative. The instructor points out that the interval of convergence for the derivative of a power series is similar to that of the original series but with a slight difference at the endpoints. By recognizing the derivative as a geometric series with a common ratio of x, the convergence is determined by the absolute value of x being less than one, thus excluding the endpoints. The interval of convergence for the derivative is from negative one to one, excluding negative one. For the integral or antiderivative of the original series, the radius of convergence remains the same, but the series converges at both endpoints, x=±1. The instructor encourages the use of the ratio test and boundary conditions to determine the interval of convergence for the integral. The summary highlights the importance of understanding how the intervals of convergence can differ at the endpoints despite having the same radius of convergence.

Mindmap
Keywords
💡Power Series
A power series is an infinite series of the form \( \sum_{n=0}^{\infty} a_n (x-c)^n \), where \( a_n \) are the coefficients and \( c \) is the center of the series. In the context of the video, power series are used to represent functions, and operations such as differentiation and integration can be performed term by term. The video discusses how to find the interval of convergence for power series, which is crucial for determining where these series can be used to represent functions.
💡Derivative
The derivative of a function measures how the function changes as its input changes. In calculus, the derivative of a function f(x) with respect to x is denoted as f'(x) or df/dx. The video explains that the derivative of a power series can be found by differentiating each term of the series, resulting in a new series where each term is the derivative of the corresponding term in the original series.
💡Integral
Integration is the process of finding a function that, when differentiated, yields another given function. The integral of a function f(x) is denoted as \( \int f(x)dx \). The video script describes integrating a power series term by term, which involves summing the integrals of each term, to find the antiderivative of the original series.
💡Term by Term
In the context of series, 'term by term' refers to the process of performing an operation on each term of the series individually. The video explains that both the derivative and integral of a power series can be found by applying the operation term by term, which simplifies the process of finding the new series after the operation.
💡Interval of Convergence
The interval of convergence (IoC) is the set of all x-values for which a power series converges. The video script discusses how to determine the IoC for both the original power series and the series obtained after differentiation and integration, noting that the IoC can differ at the endpoints.
💡Ratio Test
The ratio test is a method used to determine the convergence of an infinite series. It involves taking the limit of the absolute value of the ratio of consecutive terms of the series as n approaches infinity. The video uses the ratio test to find the IoC of the power series by examining the limit of \( |x| / (1 + 1/n) \) as n approaches infinity.
💡Harmonic Series
A harmonic series is a type of divergent infinite series where the terms are the reciprocals of the natural numbers, such as \( 1 + 1/2 + 1/3 + \ldots \). In the video, when x equals one in the power series, the series becomes a harmonic series, which is known to diverge.
💡Alternating Harmonic Series
An alternating harmonic series is a variation of the harmonic series where the terms alternate in sign, such as \( 1 - 1/2 + 1/3 - 1/4 + \ldots \). The video mentions that when x equals negative one, the series becomes an alternating harmonic series, which converges by the alternating series test.
💡Geometric Series
A geometric series is an infinite series where each term after the first is found by multiplying the previous term by a constant called the common ratio. In the video, the derivative of the power series is identified as a geometric series with a common ratio of x, and its convergence is determined by the absolute value of the common ratio being less than one.
💡Endpoint
In the context of the interval of convergence, an endpoint is a value at the boundary of the interval. The video explains that the behavior of the series at the endpoints can affect the IoC, and different series may converge or diverge at these points.
💡P-Series
A p-series is an infinite series of the form \( \sum_{n=1}^{\infty} n^{-p} \), where p is a real number. The video uses the p-series to illustrate the divergence of the series when x equals one and the convergence when the series has alternating signs and x equals negative one.
Highlights

Derivatives and integrals of power series can be taken term by term.

The derivative of a power series is the sum of the derivatives of each term.

The derivative of \( x^n \) is \( n \times x^{n-1} \).

The derivative of a power series results in \( x^{n-1} \) after canceling terms.

Integration of a power series is also done term by term.

The integral of a power series results in \( \frac{x^{n+1}}{n+1} \).

The interval of convergence for a power series is crucial for these operations.

The ratio test is used to determine the interval of convergence.

The limit of \( \frac{x^{n+1}}{n+1} \) as \( n \to \infty \) is \( |x| \).

A power series converges if \( |x| < 1 \), diverges if \( |x| > 1 \), and is inconclusive if \( |x| = 1 \).

When \( x = 1 \), the series diverges as it becomes the harmonic series.

When \( x = -1 \), the series converges as it becomes the alternating harmonic series.

The interval of convergence for the original power series is \( -1 \leq x < 1 \).

The derivative of a power series is a geometric series with a common ratio of \( x \).

The interval of convergence for the derivative is \( -1 < x < 1 \), excluding the endpoints.

The antiderivative of a power series converges at the endpoints, unlike the original series.

The interval of convergence for the antiderivative is \( -1 \leq x \leq 1 \).

The radius of convergence is the same for the original series, its derivative, and its antiderivative.

The endpoints of convergence can differ significantly for derivatives and antiderivatives.

Transcripts

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Power SeriesDerivativesIntegralsConvergenceRatio TestGeometric SeriesHarmonic SeriesAlternating SeriesMathematics EducationCalculus Techniques