Further Pure Mathematics WFM01: SERIES Topical Past Paper Questions

$$ \frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}=x^{2}y+\frac{\mathrm{d}y}{\mathrm{d}x} $$ (a) Show that $$ \frac{\mathrm{d}^{4}y}{\mathrm{d}x^{4}}=\frac{\mathrm{d}^{3}y}{\mathrm{d}x^{3}}+Ax^{2}\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}+Bx\frac{\mathrm{d}y}{\mathrm{d}x}+Cy $$ where $A$, $…

(a) Express $ \frac{2}{r(r+1)(r+2)} $ in partial fractions. (2) (b) Use the answer to part (a) and the method of differences to show that $$ \sum_{r=1}^{n}\frac{2}{r(r+1)(r+2)}=\frac{n(n+a)}{2(n+b)(n+c)} $$ where $a$, $b$ and $c$ are integers to be determined. (5) (c) Hence form…

Given that $ y = e^{x} \sin x $ (a) show that $$ \frac{\mathrm{d}^{6}y}{\mathrm{d}x^{6}}=k\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}} $$ where k is a constant to be determined. (4) (b) Hence determine the first 5 non-zero terms in the Maclaurin series expansion for y, giving each coe…