Euler transform

Definition

The Euler transform is a technique used to accelerate the convergence of an alternating series. It transforms a given series into another with potentially faster convergence.

5 Must Know Facts For Your Next Test

1. The Euler transform can be applied to any alternating series where the terms decrease in absolute value.
2. It works by creating a new series whose terms are derived from the original series' partial sums.
3. The transformed series often converges more rapidly compared to the original alternating series.
4. The general form involves taking weighted averages of partial sums of the original series.
5. The transform is particularly useful for improving numerical approximations of sums.

Review Questions

• What type of series can benefit from the Euler transform?
• How does the Euler transform affect the convergence of an alternating series?
• Describe how the terms of a new series are derived using the Euler transform.

Related terms

Alternating Series: A sequence where consecutive terms have opposite signs, usually written as $a_1 - a_2 + a_3 - a_4 + \ldots$.

Convergence: A property of a sequence or series that approaches a finite limit as more terms are added.

Partial Sum: The sum of the first $n$ terms in a sequence or series, often denoted as $S_n = \sum_{i=1}^{n}a_i$.