---
tag: A93A03C5XR
kind: definition
name: Van Wijngaarden transformation
msc: [65B10, 40A25]
defines: K3KADGATHA
uses: [017BR1M98J, 0NB643K60Q]
sources:
  - url: https://en.wikipedia.org/wiki/Van_Wijngaarden_transformation
---

## Definition

Let $\sum_{k \ge 0} (-1)^k a_k$ be an [[alternating series|017BR1M98J]] with [[partial sums|0NB643K60Q]] $s_k = \sum_{i=0}^{k} (-1)^i a_i$.
The *van Wijngaarden transformation* builds the table $(s_{j,k})_{j, k \ge 0}$ by
$$s_{0,k} = s_k, \qquad s_{j+1,k} = \frac{s_{j,k} + s_{j,k+1}}{2}.$$
The entries $s_{j,k}$ with $j \ge 1$ serve as approximations of the sum of the series.

## Remarks

By induction, $s_{j,k} = 2^{-j} \sum_{i=0}^{j} \binom{j}{i} s_{k+i}$, a binomially weighted mean of $j + 1$ consecutive partial sums.
Example: for the Leibniz series, $s_0, \dots, s_3$ are $1, 0.6667, 0.8667, 0.7238$.
The next rows are $0.8333, 0.7667, 0.7952$, then $0.8000, 0.7810$, then $0.7905$, which is within $0.006$ of $\frac{\pi}{4} \approx 0.7854$.
Non-example: the series $\sum 2^{-k}$ is not alternating, and its partial sums $1, 1.5, 1.75$ give the averages $1.25, 1.625$.
These are worse than the last partial sum; the method relies on the alternation of signs.

## Explanations

### L1 human

Write the partial sums in a row, then write the averages of neighbours below them, and repeat.