Further Mathematics 9231: Summation of series Topical Past Paper Questions

The diagram shows the curve $ y=\frac{1}{\sqrt{4x^{2}+4x+3}} $ for $ x\geqslant0 $, together with a set of n rectangles of unit width. By considering the sum of the areas of these rectangles, show that $$ \sum_{r=1}^{n}\frac{1}{\sqrt{4r^{2}+4r+3}}<\frac{1}{2}\ln\left(\frac{1+2n+\…

The curve C has parametric equations $$ x=\sinh t\quad and\quad y=t+\cosh t. $$ (a) Find $ \frac{dy}{dx} $ in terms of t. [2] (b) Show that $ \frac{d^{2}y}{dx^{2}}=\frac{1-\sinh t}{\cosh^{3}t} $. [4] (c) Find the Maclaurin's series for y in terms of x up to and including the term…

The diagram shows the curve with equation $ y=\left(\frac{1}{2}\right)^{x} $ for $ 0 \leq x \leq 1 $, together with a set of N rectangles each of width $ \frac{1}{N} $. (a) By considering the sum of the areas of these rectangles, show that $ \int_{0}^{1}\left(\frac{1}{2}\right)^{…