Sine (Maclaurin)
\(\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\)
Taylor-sarjalaskin
Luo Taylor- ja Maclaurin-sarjakehitelmiä vaihe vaiheelta polynomiapproksimaatioilla
Tulokset
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Theory & Formula
A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. When centered at x = 0, it's called a Maclaurin series.
Key properties:
- Polynomial Approximation: Taylor series approximate smooth functions using polynomials
- Convergence: Series converges within radius of convergence
- Error Term: Remainder R_n = f^(n+1)(c)(x-a)^(n+1)/(n+1)! for some c between a and x
- Applications: Used in numerical analysis, physics, engineering for approximations
- Common Series: sin(x), cos(x), e^x, ln(1+x), (1+x)^n all have simple Taylor expansions
\(f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n\)
Worked Examples
Cosine (Maclaurin)
\(\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\)
Exponential (Maclaurin)
\(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\)
Natural Log
\(\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots\)