Example of Sequence with Order of Convergence 5 to 3

[joy1819]

Homework Statement

Give an example of a sequence which converges to 3 with order of convergence 5. Prove your answer.

Homework Equations

Order of convergence: {p_n} converges to p with order $\alpha$.

lim_{n$\rightarrow\infty$} |p_n+1 - p|/|p_n-p|^$\alpha$ = $\lambda$

The Attempt at a Solution

lim_{n$\rightarrow\infty$}| p_n+1 - 3|/|p_n-3|⁵ = $\lambda$

 

[lippyka]

Let pn+1 = 3 + \frac{1}{n^5} and pn = 3 + \frac{1}{n^4}. Then, limn\rightarrow\infty|pn+1 - 3|/|pn-3|5 = limn\rightarrow\infty \frac{1/n^5}{1/n^4} 5 = limn\rightarrow\infty \frac{1}{n} 5 = 0 Therefore, the sequence {pn} converges to 3 with order of convergence 5.