# Further Maths - Taylor Series

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/taylor-series/ · Updated: 2021-12-01 · Tags: school, further-maths, further-pure-1

## See Also
- [Further Maths - Maclaurin Series](https://ollybritton.com/notes/a-level/further-maths/topics/maclaurin-series/)
- [Further Maths - Limits](https://ollybritton.com/notes/a-level/further-maths/topics/limits/)

## Flashcards
### 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 $\frac{\text{d}y}{\text{d}x}|_{x_0}$??
$$
y = y_0 + \frac{(x - x_0)}{1!} \frac{\text{d}y}{\text{d}x}|_{x_0} + \frac{(x - x_0)}{2!} \frac{\text{d}^2 y}{\text{d}x^2}|_{x_0} + \ldots
$$

### 2022-04-13
##### "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.

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