Maclaurin series of sine function

Takeaways

• 😀 The video explains how to calculate the Maclaurin series (Taylor series at x0 = 0) for the sine function.
• 😀 To calculate the Maclaurin series, several derivatives of the sine function need to be calculated first: sine, cosine, negative sine, and negative cosine.
• 😀 The derivatives of the sine function are as follows: sin(x), cos(x), -sin(x), -cos(x), and then it repeats.
• 😀 The values of the sine function and its derivatives are evaluated at x = 0. For instance, sin(0) = 0, cos(0) = 1, and so on.
• 😀 After evaluating the derivatives at x = 0, the Maclaurin series begins to take shape, with many terms being zero and others contributing to the final series.
• 😀 Terms where the derivatives are zero are omitted, leading to only the terms with odd powers of x remaining.
• 😀 The Maclaurin series for sine(x) is composed of odd powers of x with alternating signs and factorial terms in the denominator.
• 😀 The series is expressed in summation notation (sigma notation), where n starts at 0 and increases by 1 for each term.
• 😀 The alternating signs in the series can be generated using (-1)^n, ensuring that the signs alternate starting with a positive value.
• 😀 The interval of convergence of the Maclaurin series for sine(x) is determined using the ratio test. The result shows that the series converges for all real values of x, meaning the interval of convergence is (-∞, ∞).