# Further Maths - Series

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/series/ · Updated: 2020-09-14 · Tags: series, school

# Series
### Core 1
- [Further Maths - Sums of Natural Numbers](https://ollybritton.com/notes/a-level/further-maths/topics/sums-of-natural-numbers/)
- [Further Maths - Sums of Squares](https://ollybritton.com/notes/a-level/further-maths/topics/sums-of-squares/)
- [Further Maths - Sums of Cubes](https://ollybritton.com/notes/a-level/further-maths/topics/sums-of-cubes/)
- [Further Maths - Series Tips and Tricks](https://ollybritton.com/notes/a-level/further-maths/topics/series-tips-tricks/)
- [Further Maths - Induction for Series](https://ollybritton.com/notes/a-level/further-maths/topics/induction-for-series/)

### Core 2
- [Further Maths - The Method of Differences](https://ollybritton.com/notes/a-level/further-maths/topics/the-method-of-differences/)

### Flashcards
##### What is a series??
A sum of sequential terms.

##### What is the notation for series??
* Sigma notation
* e.g. $\sum^{n}_{r = 1} n$

##### How do you write $n$-th term at A-level??
$$
U_r = f(r)
$$

##### What sequence does $\sum^{n}_{r = 1} (3r - 1)$ describe??
$$
2, 5, 8, 11...
$$

##### What is $\sum^3_{r = 1} (2r)$??
$12$.

##### What is the name for a summation of a sequence??
A series.

##### $1 + 4 + 7 + 10...$ is a...??
A series.

##### $1, 4, 7, 10...$ is a...??
A sequence.

##### How can you find the sum of a series that starts at $r = k$??
$$
\sum^{n}_{r = 1} f(r) - \sum^{k-1}_{r=1} f(r)
$$

##### How can you rewrite $\sum^{n}_{r=k}$??
$$
\sum^{n}_{r=1} f(r) - \sum^{k-1}_{r=1}??
$$

##### What is $\sum^{n}_{r=1} f(r) - \sum^{k-1}_{r=1} f(r)$ equivalent to??
$$
\sum^{n}_{r=k} f(r)
$$

##### How can you rewrite $\sum^{n}_{r=1} kf(r)$??
$$
k \times \sum^{n}_{r=1} f(r)
$$

##### What's an alternate form of $k \times \sum^{n}_{r=1} f(r)$??
$$
\sum^{n}_{r=1} kf(r)
$$

##### How could you rewrite $\sum^{n}_{r = 1} (f(r) + g(r))$??
$$
\sum^{n}_{r=1} f(r) + \sum^{n}_{r=1} g(r)
$$

##### What is $\sum^{n}_{r=1} k$ the same as??
$$
k \times n
$$

##### How could you rewrite $\sum^{25}_{r=1} (3r + 1)$??
$$
3 \sum^{25}_{r=1} r + n
$$

##### How can you find the sum of a series that starts at $k$, not $1$??
$$
\sum^{n}_{r=k} f(r) = \sum^{n}_{r=1} f(r) - \sum^{k-1}_{r=1} f(r)
$$

##### What's another way of writing $\sum^{n}_{r=k}$??
$$
\sum^{n}_{r = 1} f(r) - \sum^{k - 1}_{r = 1} f(r)
$$

##### How do you deal with something other than $n$ at the top of the $\Sigma$, like $\sum^{4n-1}_{r=1}$??
Instead of substituting $n$, you subsititue $4n-1$ into the formula.

##### What's the value of $\sum^{2n}_{r=1} 5$??
$$
10n
$$

##### If you show $\sum^{4n-1}_{r=1} (3r+1) = 24n^2 - 2n - 1$, what's the first step to solving $\sum^{7}_{r=1} (3r+1)$??
First solve:

$$
4n - 1 = 7
4n = 8
n = 2
$$

##### If $\sum^{n}_{r=1}$ is linear, the expression for $\sum^{n}_{r=1} r$ is...??
Quadratic.

##### If $\sum^{n}_{r=1}$ is linear, the expression for $\sum^{n}_{r=1} r^2$ is...??
Cubic.

##### If $\sum^{n}_{r=1}$ is linear, the expression for $\sum^{n}_{r=1} r^3$ is...??
Quartic.

##### How could you simplify $\frac{1}{6}n(n+1)(2n+2)$??
$$
\frac{1}{3}n(n+1)^2
$$

---
Olly Britton — https://ollybritton.com. Machine-readable index: https://ollybritton.com/llms.txt