# Further Maths - The Method of Differences

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/the-method-of-differences/ · Updated: 2021-01-26 · Tags: further-maths, series

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

### Flashcards
##### What is the requirement of a series for the method of differences to be applicable??
The general term, $u_r$ of a series can be expressed in the form $f(r) - f(r+1)$.

##### If the general term of a series $u_r$ can be expressed as $f(r) - f(r + 1)$, how could you write the series??
$$
\sum^n\_{r = 1} (f(r) - f(r + 1))
$$

##### For a general series $\sum^n_{r = 1} (f(r) - f(r + 1))$, what is $u_1$??
$$
f(1) - f(2)
$$

##### For a general series $\sum^n_{r = 1} (f(r) - f(r + 1))$, what is $u_2$??
$$
f(2) - f(3)
$$

##### For a general series $\sum^n_{r = 1} (f(r) - f(r + 1))$, what is $u_r$??
$$
f(r) - f(r + 1)
$$

##### For a general series $\sum^n_{r = 1} (f(r) - f(r + 1))$, what are the first and last few terms of the series??
$$
\begin{align*}
+\\ f(1) &- f(2) \\\\
+\\ f(2) &- f(3) \\\\
+\\ f(3) &- f(4) \\\\
\dots \\\\
+\\ f(n) &- f(n + 1) \\
\end{align*}
$$

##### $$\begin{align*} +\ f(1) &- f(2) \\ +\\ f(2) &- f(3) \\ +\ f(3) &- f(4) \\ \dots \\ +\ f(n) &- f(n + 1) \end{align*}$$ What does this cancel down to??
$$
f(1) - f(n + 1)
$$

#### 2021-01-28
##### If you're doing a method of differences question with three partial fractions, what is probably true that means they cancel out??
The fractions along the diagonals add up to something that is cancelled out.

##### $$\frac{1}{3} - \frac{1}{2(n + 1)}$$ What is the value of this expression as $n \to \infty$??
$$
\frac{1}{3}
$$

##### If asked to find the limit of a series after a Method of Differences question, should you combine the fractions or leave them seperated??
Leave them seperated.

##### $$\frac{1}{2(n + 1)}$$ How would you write what this is equal to in an exam??
As $n \to \infty$

$$
\frac{1}{2(n+1)} \to 0
$$

### 2022-05-15
##### $$\frac{1}{1} + \frac{1}{3} - \frac{1}{2}$$ $$\frac{1}{2} + \frac{1}{6} - \frac{1}{4}$$ $$\frac{1}{3} + \frac{1}{9} - \frac{1}{8}$$ How would you notice these fractions cancelling out for the method of differences??
They cancel along the diagonal.

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