Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The equation $x = \frac{10}{e^{2x} - 1}$ has a single positive real root, called $\alpha$.
(a(i))[2]
Demonstrate that $\alpha$ lies in the interval from $x = 1$ to $x = 2$.
(a(ii))[2]
Show that, if the sequence of positive terms defined by the iterative formula $x_{n+1} = \frac{1}{2} \ln\left(1 + \frac{10}{x_n}\right)$ converges, then its limit is $\alpha$.
(a(iii))[3]
Use the iterative formula to find $\alpha$ correct to $2$ decimal places. Show every iterate to $4$ d.p.
Worked solution & mark scheme
This 7-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Check the sign of $\frac{x-10}{e^{2x}-1}$ at $x=1$ and $x=2$” …