Method of differences with partial fractionsAQA A-Level Further Maths: Flashcards

What these 12 flashcards ask

What is the method of differences?
\sum{r=1}^{n}[f(r)-f(r+1)]= ?
\sum{r=1}^{n}[f(r+1)-f(r)]= ?
Partial fractions of \frac{1}{r(r+1)}?
Partial fractions of \frac{1}{r(r+2)}?
Partial fractions of \frac{1}{(2r-1)(2r+1)}?
\sum{r=1}^{n}\frac{1}{r(r+1)}?
Which terms survive for f(r)-f(r+2)?
How do you find \sum{r=a}^{b}ur from a formula for Sn?
How do you get a sum to infinity from Sn?
\sum{r=1}^{n}\left[r^3-(r-1)^3\right]?
A quick check on a telescoped formula?

Exam questions on Method of differences with partial fractions

Let $S_n=\sum_{r=1}^{n}\frac{1}{r(r+1)}$ .
Find the least value of $n$ for which $S_n>0.99$ .2 marks
Let $u_r=r^3-(r-1)^3$ for positive integers $r$ .
Hence find $\sum_{r=11}^{20}\left(3r^2-3r+1\right)$ .2 marks
Let $u_r=\frac{1}{(2r-1)(2r+1)}$ and $S_n=\sum_{r=1}^{n}u_r$ .
Express $u_r$ in partial fractions and hence show that $S_n=\frac{n}{2n+1}$ .3 marks

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).