Hints to Homework Assignments of MATH 450 Fall 2008
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Problem (9) in Page 25. We have found the limit of the sequence is − using l’Hospital
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rule. The rate of convergence used here is defined in the similar manner as that in the
problem (8) in the same page. Specifically,
Let lim∗ f (x) = a. If
x→x
|f (x) − a| = O(|x − x∗ |p )
as x → x∗ , the we say that the rate of f (x) converging to a is p.
For example, the rate of convergence of f (x) = 1 + sin(x3 ) approaching 1 as x → 0 is 3,
because
|f (x) − 1| = |1 + sin(x3 ) − 1|
= | sin(x3 )|
= sin(|x|3 )
= O(|x|3 ).
To find the rate of convergence of
f (h) =
1
[(1 + h) − eh ]
h2
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approaching − as h → 0, we assume that the rate is p. But then the problem becomes
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for which p
1
|f (h) − (− )| = O(hp )
2
as h → 0. Using the definition of big O you shall be able to find the right choice of p.
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