Further Maths - Series

Created: September 14, 2020

Series

Core 1

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 $\frac{1}{6}n(n+1)(2n+2)$?
\[\frac{1}{3}n(n+1)^2\]


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