Approximaths

Euler summation I

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So far we’ve seen Padé’s approximants. These have enabled us to approximate a function from its corresponding Taylor series and then transform this (potentially non-convergent) series into a convergent rational fraction.

We’d like to introduce other techniques that could be used to sum non-convergent series. First, we’ll take a look at Euler summation.

If the series \sum_{n=0}^{\infty} a_n is algebraically divergent (the terms blow up like some power of n), then the series:

\displaystyle f(x) = \sum_{n=0}^{\infty}a_n x^n

converges for x \in (-1,1). If the limit

\displaystyle E := \lim_{x \to 1_{-}}f(x)

exists and is finite then it is called the Euler sum E of the original series.

For example, consider the divergent series:

\displaystyle 1 - 1 + 1 - 1 + 1 - 1 + ...

and multiply each n-term with x^n (starting at n = 0):

\displaystyle f(x) = x^0 - x^1 + x^2 - x^3 + x^4 - x^5 + ...
\displaystyle = \frac{1}{1-(-x)} = \frac{1}{1+x}
\displaystyle E = \lim_{x \to 1_{-}}{\frac{1}{1+x}} = \frac{1}{2}

Therefore (according to Euler summation):

\displaystyle 1 - 1 + 1 - 1 + 1 - 1 + ... = \frac{1}{2}

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