Calculus How To

Asymptotic Error (Rate of Convergence)

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The asymptotic error constant (λ) tells you something about the behavior of a sequence’s errors: differences between the terms of the sequence and the sequence’s limit. The asymptotic error constant affects the speed of convergence in conjunction with the order of convergence.

How to Find the Asymptotic Error Constant

You can find a sequence’s asymptotic error constant λ with the following limit definition:
asymptotic error constant

Where:

  • α = order of convergence,
  • pn = a sequence,
  • p = where the sequence converges (pn ≠ p).

As the number of terms increases, the sequence approaches the limit or horizontal asymptote; λ tells you the rate at which that happens. If positive constants λ and α exist for the above limit, then the sequence converges to p of order of α at rate λ.

In general, a sequence with a higher order of convergence will converge faster than a sequence with a lower order of convergence (the asymptotic constant also affects the speed of convergence, but not to the extent that the order does)[1]:

  • If α = 1, pn converges to p linearly.
  • If α = 2, pn converges to p quadratically.
  • If α = 3, pn converges to p cubically.
  • If 0 < α < 1, pn converges to p suberlinearly.
  • If 1 < α < 2, pn converges to p superlinearly.

While there are many possible values for order of convergence, most of the sequences you’ll come across in beginning calculus and analysis classes will have α = 1 or α = 2.

For example, the following two sequences both converge to 5:
order of convergence example

However, they converge with different orders of convergence.


The first sequence converges linearly (α = 1) to 5:
order of convergence example 1

The second sequence converges quadratically (α = 2) to 5:
order of convergence example 2

Asymptotic Error Constant: Example

Example question: What is the asymptotic error constant for the sequence:
find the asymptotic error constant example

Step 1: Find the limit of the sequence:
step 1 example 1


For this sequence, p = 0.

Step 2: Insert p from Step 1 and α from Step 2 into the formula and solve. Using the definition of the asymptotic error constant, the sequence has order of convergence α if λ exists in the limit and is finite. This particular sequence converges linearly (α = 1):
asymptotic error example solution 1a


References

Chapter 2: Solutions of Equations in One Variable. Retrieved August 14, 2021 from: https://www.math.tamu.edu/~smpun/MATH417/Chapter-2.pdf

CITE THIS AS:
Stephanie Glen. "Asymptotic Error (Rate of Convergence)" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/asymptotic-error-rate-of-convergence/
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