Further Maths - Taylor Series

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.

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 $\frac{\text{d}y}{\text{d}x} \vert _ {x _ 0}$?

\[y = y_0 + \frac{(x - x_0)}{1!} \frac{\text{d}y}{\text{d}x}|_{x_0} + \frac{(x - x_0)}{2!} \frac{\text{d}^2 y}{\text{d}x^2}|_{x_0} + \ldots\]

“Find the Taylor series about $x = 0$ of…”

\[\ln\left( \frac{1 + 2x}{(1 - 2x)^2} \right)\]

Where’s one place you could go wrong here?

Adding together the individual expansions rather than subtracting.

See Also