Approximaths

Euler summation II

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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.

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