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