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2021-12-01
What is the Maclaurin series a special case of??
The Taylor series.
What is the formula for the Taylor series about $x = a$??
$$ f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + … $$
When is the Taylor series valid for $x = a$??
When $f^{(n)}(a)$ exists and is finite for all natural numbers and for values of $x$ for which the infinite series converges.
What is the formula for the Taylor series for $f(x + a)$??
$$ f(x + a) = f(a) + f'(a)x + \frac{f''(a)}{2!}x^2 + \frac{f'''(a)}{3!}x^3 + … $$
How can you derive the Taylor series??
Considering the Maclaurin expansion for $g(x)$ where $g(x) = f(x + a)$.
Instead of finding the Nth derivative of $f(x) = e^{x}\sin(x)$ for the Taylor expansion, how can you find it much quicker??
Multiply the series for each part together individually.
2022-02-17
How can you transform the Maclaurin series expansion $$f(x) = f(0) + x f'(0) + \frac{x^2}{2!} f''(0) \ldots$$ to the series solution of the differential equation $$f(x, y) = \frac{\text{d}y}{\text{d}x}$$ in terms of $x_0$, $y_0$ and $\frac{\text{d}y}{\text{d}x}|_{x_0}$??
$$ y = y_0 + \frac{(x - x_0)}{1!} \frac{\text{d}y}{\text{d}x}|{x_0} + \frac{(x - x_0)}{2!} \frac{\text{d}^2 y}{\text{d}x^2}|{x_0} + \ldots $$
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date: 2021-12-01 16:00
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title: Further Maths - Taylor Series