Worked example: cosine function from power series | Series | AP Calculus BC | Khan Academy

Khan Academy
13 Oct 201406:08
EducationalLearning
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TLDRIn this educational video, the presenter tackles the evaluation of an infinite power series with a peculiar twist: the series involves powers of x to the sixth multiplied by a trigonometric hint, pi over two. The presenter encourages viewers to pause and attempt the problem, hinting at the Maclaurin Series for cosine as a potential solution. Through a step-by-step expansion of the series and a clever substitution of x with x to the third, the presenter reveals that the series is, in fact, the cosine of x cubed. The final evaluation of the series at x as the cube root of pi over two results in a surprising and satisfying conclusion: the series sums to zero.

Takeaways
  • πŸ“š The problem involves evaluating an infinite series with a specific power of x.
  • πŸ” The series is expressed as the sum from n=0 to infinity of (-1)^n * x^(6n) / (2n)!.
  • πŸ€” The challenge is to determine what function this power series represents.
  • πŸ‘€ The hint suggests that the series might be related to a trigonometric function, given the presence of pi/2.
  • πŸ“ˆ The series is expanded to show its terms, revealing a pattern similar to the Maclaurin series for cosine.
  • 🧩 The series can be manipulated by replacing x with x^(1/3) to match the form of the cosine series.
  • πŸ”„ This manipulation leads to the series representing the cosine of x^(1/3).
  • πŸ“‰ The goal is to evaluate the series when x equals the cubed root of pi/2.
  • 🎯 By substituting x with the cubed root of pi/2, the series simplifies to the cosine of pi/2.
  • πŸ”š The final evaluation shows that the cosine of pi/2 is zero, concluding the problem.
Q & A

  • What is the goal of the video?

    -The goal of the video is to evaluate the given power series when x is equal to the cubed root of pi over two.

  • What is the hint provided in the video for solving the problem?

    -The hint is to figure out what function the power series represents and then use that function to evaluate the series. Additionally, the number pi over two suggests using a trigonometric function.

  • What is the power series given in the video?

    -The power series is the sum from n equals zero to infinity of (-1)^n * x^(6n) / (2^n * n!).

  • What is the expansion of the power series for the first few terms?

    -The expansion starts with 1 (when n=0), then -x^6 / 2! (when n=1), +x^12 / 4! (when n=2), and -x^18 / 6! (when n=3), continuing in this pattern.

  • What trigonometric function does the power series resemble?

    -The power series resembles the Maclaurin series for the cosine function.

  • What is the Maclaurin series for cosine of x?

    -The Maclaurin series for cosine of x is approximately equal to 1 - x^2 / 2! + x^4 / 4! - x^6 / 6! and continues with alternating signs and increasing powers of x.

  • How can the power series be related to the cosine function?

    -By replacing x with x^(1/3) in the cosine function, the power series matches the given series, indicating that the series is the Maclaurin series for cos(x^(1/3)).

  • What is the significance of the number pi over two in the context of the problem?

    -The number pi over two is used to evaluate the power series when x is the cubed root of pi over two, which simplifies the evaluation process.

  • How does taking the cube of the cubed root of pi over two simplify the problem?

    -Taking the cube of the cubed root of pi over two results in pi over two, which is a known value for the cosine function, making the evaluation straightforward.

  • What is the final evaluation of the power series when x is the cubed root of pi over two?

    -The final evaluation is cos(pi/2), which is equal to zero.

Outlines
00:00
πŸ“š Evaluating an Infinite Power Series

The speaker introduces an infinite power series involving a complex exponential term and aims to evaluate it when x equals the cubed root of pi over two. They encourage the audience to try solving it before revealing the method. The series is expanded to show its terms and compared to the Maclaurin series for cosine, suggesting a trigonometric function might be involved. The hint given is to identify the function represented by the series and then use that to evaluate it, with a particular focus on the 'suspicious' number pi over two, which might be related to a trigonometric evaluation.

05:01
🧩 Connecting the Series to Cosine Function

The speaker identifies the power series as being closely related to the Maclaurin series for the cosine function, with the exponents of x needing adjustment to match. They propose replacing x with x to the power of 1/3 to align with the cosine series, leading to the series representing the cosine of x cubed. The evaluation of the series when x is the cubed root of pi over two simplifies to the cosine of pi over two, which is zero. This clever manipulation of the series and the use of trigonometric identities provide a straightforward solution to the problem.

Mindmap
Keywords
πŸ’‘Infinite Series
An infinite series is the sum of the terms of an infinite sequence. In the context of the video, the series is defined by a formula involving powers of -1 and x, and factorials. The series is central to the video's theme as it is the mathematical object the presenter is attempting to evaluate.
πŸ’‘Power Series
A power series is a series in which each term is a power of x, multiplied by a coefficient. The video script discusses evaluating a specific power series when x is a certain value, which is a common technique in calculus for representing functions as series.
πŸ’‘Cubed Root
The cubed root of a number is a value that, when multiplied by itself three times, gives the original number. In the video, the cubed root of pi over two is used as the value for x in the power series, which is a key step in evaluating the series.
πŸ’‘Factorial
A factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. In the script, factorials are used in the denominator of the series terms, indicating the importance of combinatorial mathematics in the evaluation of the series.
πŸ’‘Trigonometric Function
Trigonometric functions relate angles to the ratios of the sides of a right triangle. The video hints at using a trigonometric function to evaluate the series, specifically the cosine function, which is closely related to the series in question.
πŸ’‘Maclaurin Series
A Maclaurin series is a type of power series used to represent a function as an infinite sum of terms, derived from the Taylor series expansion around zero. The script mentions the Maclaurin series for cosine, which is used as a reference point to identify the series being evaluated.
πŸ’‘Cosine Function
The cosine function is one of the fundamental trigonometric functions, representing the horizontal coordinate of a point on the unit circle. The video script relates the given series to the cosine function, suggesting that the series is a representation of the cosine of a certain power of x.
πŸ’‘Exponents
Exponents are used to denote repeated multiplication of a number by itself. In the script, the exponents on x in the series terms are manipulated to match the form of the cosine function's Maclaurin series, which is crucial for identifying the function represented by the series.
πŸ’‘Evaluation
Evaluation in the context of the video refers to the process of finding the sum of the series for a specific value of x. The script's goal is to evaluate the given power series when x is the cubed root of pi over two, which involves understanding the relationship between the series and trigonometric functions.
πŸ’‘Zero
In mathematics, zero is the integer that represents the absence of quantity. The video concludes with the evaluation of the series resulting in zero, which is a significant outcome given the specific value of x used in the series.
Highlights

The presenter introduces an infinite series to be evaluated for a specific value of x.

The series involves powers of -1, x to the sixth power, and factorials in the denominator.

The goal is to evaluate the series when x equals the cubed root of pi over two.

The audience is encouraged to pause and attempt the problem independently.

A hint is given to determine the function that the power series represents.

The number pi over two is highlighted as a clue for potential trigonometric function usage.

The presenter expands the power series to get a better understanding of its structure.

The expansion reveals a pattern similar to the Maclaurin series for cosine.

The series is compared to the cosine Maclaurin series term by term.

A realization is made that the series could represent cosine of x cubed.

The presenter suggests replacing x with x to the third power to match the cosine series.

The series is rewritten as the cosine of x to the third power.

The evaluation of the series at the cubed root of pi over two is discussed.

The cube of the cubed root is calculated, simplifying the trigonometric function.

The final evaluation results in the cosine of pi over two, which is zero.

The problem is concluded with a clear and concise solution.

Transcripts

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Infinite SeriesTrigonometryCosine FunctionPower SeriesMaclaurin SeriesCubed RootPi Over TwoMathematics TutorialEducational ContentMath Analysis