MATH661

Approximation Techniques - Exercises

1.Rate and order of convergence

Exercise 1. Estimate the rate and order of convergence of a sequence from the samples

1, \frac{0.9}{2}, \frac{0.81}{4}, \frac{0.729}{8} .

Exercise 2. Construct a convergence plot for the sequence

a_{n} = \frac{1}{\ln n} .

Can you estimate the rate and order of convergence?

Exercise 3. Construct a convergence plot for the sequence

a_{n} = e^{- n} .

Can you estimate the rate and order of convergence?

Exercise 4. Plot in logarithmic coordinates $a_{n} = \sqrt{n}$ , $a_{n + 1} - a_{n}$ , $a_{n + 2} - a_{n}$ , $\dots$ , $a_{n + 10} - a_{n}$ . Do the $a_{n + k} - a_{n}$ sequences converge?

2.Aitken acceleration

Exercise 5. What is the order of convergence of the sequence $x_{n} = 1 / n$ ? Construct a convergence plot.

Exercise 6. Supperimpose on your previous plot the convergence plot of Aitken acceleration applied to $x_{n} = 1 / n$ . What order of convergence is achieved?

Exercise 7. What is the order of convergence of the sequence $x_{n} = 1 / n^{2}$ ? Construct a convergence plot.

Exercise 8. Supperimpose on your previous plot the convergence plot of Aitken acceleration applied to $x_{n} = 1 / n^{2}$ . What order of convergence is achieved?

3.Approximation techniques

Exercise 9. Find six significant digits of $\sqrt{2}$ through the continued fraction

\sqrt{2} = 1 + \overset{\infty}{\underset{k = 1}{K}} \frac{1}{2}

Exercise 10. Consider the sequence $a_{1} = \sqrt{2}$ , $a_{n + 1} = \sqrt{2 + a_{n}}, n \in ℕ \ {0}$ . Express $a_{n}$ through function composition

a_{n} = T_{n} (z), T_{n} = t_{0} \circ t_{1} \circ \dots \circ t_{n} .

Exercise 11. Express the Wallis product through function composition.

Exercise 12. Find the order of convergence of the Ramanujan series.