Polynomial approximation of functions (part 6) | Summary and Q&A

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April 30, 2008
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Polynomial approximation of functions (part 6)

TL;DR

The Maclaurin Series representations of e to the x, cosine of x, and sine of x exhibit a striking similarity and connection, with the polynomial representation of e to the x being almost identical to the addition of the polynomial representations of cosine and sine, except for a few sign changes.

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Questions & Answers

Q: Why does the Maclaurin Series representation of e to the x equal 1 + x + x^2/2! + x^3/3! + ...?

The Maclaurin Series for e to the x is derived using calculus and the properties of exponentials. Each term in the series represents a power of x divided by the factorial of the corresponding power.

Q: What is the connection between the Maclaurin Series representations of cosine and sine?

The Maclaurin Series representations of cosine and sine exhibit a similar pattern, with alternating positive and negative signs. These series can be derived from the unit circle definition of cosine and sine, as well as the properties of power series.

Q: How does the addition of the polynomial representations of cosine and sine relate to e to the x?

When the polynomial representations of cosine and sine are added together, the resulting series is almost identical to the polynomial representation of e to the x, except for a few sign changes. This shows a remarkable connection between trigonometric functions and exponential functions.

Q: What is the significance of the connection between the Maclaurin Series representations of e to the x, cosine of x, and sine of x?

The connection between these series highlights a hidden order in the universe, suggesting a deeper relationship between trigonometry and exponential growth. This relationship has been studied for centuries and continues to fascinate mathematicians.

Summary & Key Takeaways

  • The Maclaurin Series representation of e to the x is 1 + x + x^2/2! + x^3/3! + ...

  • The Maclaurin Series representation of cosine of x is 1 - x^2/2! + x^4/4! - ...

  • The Maclaurin Series representation of sine of x is x - x^3/3! + x^5/5! - ...

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