Partial fractions of \frac{1}{r(r+2)}?

\frac12\left(\frac1r-\frac1{r+2}\right)

How many terms survive for a gap of two, such as \frac1r-\frac1{r+2}?

Four: the first two and the last two.

Partial fractions of \frac{1}{(2r-1)(2r+1)}?

\frac12\left(\frac1{2r-1}-\frac1{2r+1}\right)

How do you find a sum to infinity from Sn?

Let n\to\infty; the leftover terms such as \frac1{n+1} tend to 0.

\frac{1}{r(r+1)(r+2)} as a difference?

\frac12\left[\frac{1}{r(r+1)}-\frac{1}{(r+1)(r+2)}\right]

The method of differencesEdexcel International A Level Further Maths: Flashcards

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What is the method of differences?

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What is the method of differences?: Writing $u_r=f(r)-f(r+1)$ so that the sum of the terms telescopes.
$\sum_{r=1}^{n}[f(r)-f(r+1)]$ ?: $f(1)-f(n+1)$
$\sum_{r=1}^{n}[f(r+1)-f(r)]$ ?: $f(n+1)-f(1)$
Partial fractions of $\frac{1}{r(r+1)}$ ?: $\frac1r-\frac1{r+1}$
Partial fractions of $\frac{1}{r(r+2)}$ ?: $\frac12\left(\frac1r-\frac1{r+2}\right)$
$\sum_{r=1}^{n}\frac{1}{r(r+1)}$ ?: $1-\frac{1}{n+1}=\frac{n}{n+1}$
How many terms survive for a gap of two, such as $\frac1r-\frac1{r+2}$ ?: Four: the first two and the last two.
Partial fractions of $\frac{1}{(2r-1)(2r+1)}$ ?: $\frac12\left(\frac1{2r-1}-\frac1{2r+1}\right)$
$\sum_{r=1}^{n}\frac{1}{(2r-1)(2r+1)}$ ?: $\frac{n}{2n+1}$
Simplify $(r+1)!-r!$ .: $r\cdot r!$
$\sum_{r=1}^{n}r\cdot r!$ ?: $(n+1)!-1$
How do you find a sum to infinity from $S_n$ ?: Let $n\to\infty$ ; the leftover terms such as $\frac1{n+1}$ tend to $0$ .
$\frac{1}{r(r+1)(r+2)}$ as a difference?: $\frac12\left[\frac{1}{r(r+1)}-\frac{1}{(r+1)(r+2)}\right]$

Exam questions on The method of differences

The general term of a series is $u_r=\frac{1}{r(r+2)}$ for $r\ge1$ .
Find $\sum_{r=1}^{\infty}u_r$ .2 marks
Let $f(r)=r!$ for positive integers $r$ .
Hence find $\sum_{r=3}^{5}r\cdot r!$ .2 marks
A series has general term $u_r=\frac{1}{(2r-1)(2r+1)}$ for $r\ge1$ .
Express $u_r$ in partial fractions.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).