Numerical Analysis - show something about the rate of convergence We are given an iterative method

Agostarawz 2022-06-29 Answered
Numerical Analysis - show something about the rate of convergence
We are given an iterative method for finding roots, x n + 1 = g ( x n ), we are given the rate of convergence of this method is p, and also that:
lim n | e n + 1 | | e n | p = c
where e n = | x x n | (I'm assuming that x is the value to which x k is converging)
Show that
lim n l o g | e n + 1 | l o g | e n | = p
Here's what I did:
lim n | e n + 1 | | e n | p = c implies
l o g ( lim n | e n + 1 | | e n | p ) = l o g ( c )
I assumed that l o g ( lim n | e n + 1 | | e n | p ) = lim n l o g ( | e n + 1 | | e n | p ) is this true? if so why?, but suppose it is.
then
lim n l o g ( | e n + 1 | | e n | p ) = l o g ( c ) implies that
p = lim n l o g | e n + 1 | l o g ( c ) l o g | e n |
And here I am stuck. Unless l o g ( c ) = 0 then I did not solve the question.
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Answers (1)

Jenna Farmer
Answered 2022-06-30 Author has 17 answers
Use that for any c > ε > 0 and n n ε large enough
c ε < | e n + 1 | | e n | p < c + ε
to get useful inequalities.
And use lim n ln ( | e n | ) =
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