Math Problem Statement
Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] f(x) = sin(x), a = 𝜋
Solution
To find the Taylor series for centered at , we need to calculate the derivatives of at and then use the Taylor series formula:
For , we first determine the derivatives:
- (and this cycle repeats every four derivatives)
Next, evaluate these derivatives at :
- (and this cycle repeats every four derivatives)
Now, we can write the Taylor series expansion at :
Substituting the evaluated derivatives:
Simplifying, we get:
Thus, the Taylor series for centered at is:
Or, more compactly, considering only the terms where is odd:
Do you want details or have any questions?
Further Questions:
- How is the Taylor series used to approximate functions near a point?
- What are the convergence properties of the Taylor series for at different points?
- How would the Taylor series change if centered at instead of ?
- Can you derive the Taylor series for centered at ?
- How can we use the Taylor series to approximate near ?
- What is the radius of convergence for the Taylor series of ?
- How would the series look if we included the remainder term ?
- What are some practical applications of Taylor series in physics and engineering?
Tip:
Remember that the Taylor series is a powerful tool for approximating functions, especially when dealing with non-linear functions and evaluating them near specific points.
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Math Problem Analysis
Mathematical Concepts
Calculus
Taylor Series
Trigonometric Functions
Formulas
Taylor series expansion formula
Sine function derivative formulas
Theorems
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Suitable Grade Level
Advanced High School or College