Series
Core 1
- [[Further Maths - Sums of Natural Numbers]]S
- [[Further Maths - Sums of Squares]]S
- [[Further Maths - Sums of Cubes]]S
- [[Further Maths - Series Tips and Tricks]]S
- [[Further Maths - Induction for Series]]S
Core 2
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 $\frac{1}{6}n(n+1)(2n+2)$??
$$ \frac{1}{3}n(n+1)^2 $$
Backlinks
- [[Further Maths - The Method of Differences]]S
- [[Further Maths - Series Tips and Tricks]]S
- [[Further Maths - Syllabus]]S
- [[Maths - Recurrence Relations]]S
Metadata
date: 2020-09-14 10:29
page: 43
tags:
- '@further-maths'
- '@?series'
- '@?school'
- '@?public'
textbook: pfmy1
title: Further Maths - Series