# Rates of Convergence of Powers

1. Suppose that 0 < q < p and that alpha_n = alpha + O(n^-p). Show that alpha_n = alpha + O(n^-q).

2. Make a table listing h, h^2, h^3, and h^4 for h = 0.5, 0.1, 0.01, and 0.001, and discuss the varying rates of convergence of these powers of h as h approaches zero.

Please provide a brief text description explaining the steps taken to solve each problem.

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#### Solution Preview

1. Suppose that 0 < q < p and that alpha_n = alpha + O(n^-p). Show that alpha_n = alpha + O(n^-q).

Proof: By definition of the big O notation, there exist positive constants N and M such that for all n > N, we have

|alpha_n - alpha| < Mn^-p. But if 0 < q < p, it follows that n^q < n^p, whence n^-q > n^-p, for all n > 1. Thus we have |alpha_n ...

#### Solution Summary

This solution investigate the rates of convergence of various powers of h as h approaches zero.