Approximaths

Category: Summation techniques

  • Euler summation II

    In the previous post, we introduced Euler summation. The following are two examples where it fails to produce a finite result.

    Consider the divergent series:

    \displaystyle 0 + 1 + 2 + 3 + 4 + \dots

    Define:

    \displaystyle \begin{array}{rcl} f(x) &=& 0x^{0} + 1x^{1} + 2x^{2} + 3x^{3} + 4x^{4} + \dots \\ &=& \sum_{n=0}^{\infty} nx^{n} = \frac{x}{(1-x)^{2}} \\ \end{array}

    The Euler sum is:

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

    Now consider the divergent series:

    \displaystyle 1 + 4 + 9 + 16 + 25 + 36 + \dots

    Define:

    \displaystyle \begin{array}{rcl} f(x) &=& 1^{2}x^{1}+2^{2}x^{2}+3^{2}x^{3}+4^{2}x^{4}+5^{2}x^{5}+6^{2}x^{6}+\dots \\ &=& \sum_{n=1}^{\infty} n^{2} x^{n} = \frac{x(1+x)}{(1-x)^{3}} \\ \end{array}

    The Euler sum is:

    \displaystyle E = \lim_{x \to 1_{-}}\frac{x(1+x)}{(1-x)^{3}} = \infty
    \displaystyle E(1^{2} + 2^{2} + 3^{2} + 4^{2} + \dots) = \infty

    Advantages

    Regularization of slowly divergent series:
    Euler summation can assign a finite value to some divergent series that oscillate or diverge slowly, such as

    \displaystyle 1 - 1 + 1 - 1 + \dots

    where E(series) = \tfrac{1}{2}.

    Improved convergence:
    For many convergent series, Euler transformation accelerates convergence, making it useful for numerical computations.

    Analytic continuation link:
    It provides a bridge between ordinary summation and more advanced summation methods (e.g. Borel or zeta regularization).

    Disadvantages

    Limited domain of applicability:
    Euler summation fails for series that diverge too rapidly, such as

    \displaystyle 1 + 2 + 3 + 4 + \dots

    where E(series) = \infty.

    Not uniquely defined for all divergent series:
    Some series cannot be assigned a finite Euler sum, or the method may yield inconsistent results depending on the transformation order.

    Weaker than analytic regularization:
    Compared to zeta or Borel summation, Euler’s method handles fewer classes of divergent series and lacks a rigorous analytic continuation framework.

  • Euler summation I

    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}