Limits: rate of convergence of Euler

In summary, the problem is to prove that the rate of convergence for Euler's method in solving y'=y is comparable to 1/n. This can be done by showing that the limit of e-(1+1/n)^n divided by 1/n is equal to e/2. This can be achieved by using Maclaurin expansion for ln(1+t) and the series expansion for e. The hint of using L'Hopital's Rule can be disregarded as it is not necessary.
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Homework Statement


Prove that the rate of convergence for Euler's method (in solving the problem y'=y) is comparable to 1/n by showing that [tex]lim_{n\rightarrow infinity} \frac{e-(1+\frac{1}{n})^n}{1/n}[/tex].


Homework Equations


Hint: Use L'Hopital's Rule
If lim x-> infinity f(x)/g(x) is in indeterminate form, then it can be evaluated as lim x-> infinity f'(x)/g'(x).
Hint: Use Maclaurin expansion for ln(1+t).
ln(1+t) = t - t^2/2 + t^3/3 - t^4/4 + t^5/5 ...


The Attempt at a Solution



L'Hopital's Rule:
Differentiating gives
[tex]lim_{n\rightarrow infinity} \frac{\frac{1}{n+1} - ln(1 + \frac{1}{n})(\frac{n+1}{n})^n}{-1/n^2}[/tex]

[tex] ln (1+1/n) = 1/n - 1/(2n^2) + 1/(3n^3) - 1/(4n^4) + 1/(5n^5) ... [/tex]

Not sure where to go from here. When I plug in the series, what can I do? I tried multiplying the fraction by [tex]\frac{-n^2}{-n^2}[/tex] but I don't think that gets me anywhere. I could apply L'Hopital again, but that seems like a monstrous function to differentiate. Any ideas?
 
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  • #2
I would say don't use l'Hopital if you are going to use series expansions anyway. (1+1/n)^n=exp(n*ln(1+1/n)). Use the series expansion on n*ln(1+1/n). Only keep 2 terms. Factor out the e. Now series expand exp. Only keep the terms that are going lead to terms with a power of n less than two.
 
  • #3
Great! that works out to e/2. Thanks for the help! I wonder why they hinted at L'hopital though.
 

What is the Euler method and how does it relate to limits?

The Euler method is a numerical method used to approximate the solutions of differential equations. It is based on the concept of breaking down the continuous function into smaller intervals and approximating the value of the function at each interval. The relation to limits comes in when we take the limit of the intervals approaching zero, which gives us a more accurate approximation.

How do you calculate the rate of convergence of the Euler method?

The rate of convergence of the Euler method can be calculated by taking the limit of the error term as the step size approaches zero. This can be done using the Big O notation, where the error term is expressed in terms of the step size, and the rate of convergence is the power to which the step size is raised.

What factors affect the rate of convergence of the Euler method?

The rate of convergence of the Euler method is affected by the step size, the order of the differential equation, and the smoothness of the function being approximated. A smaller step size and a higher order differential equation result in a faster convergence rate, while a less smooth function can lead to a slower convergence rate.

How does the rate of convergence of the Euler method compare to other numerical methods?

The rate of convergence of the Euler method is generally slower than other numerical methods, such as the Runge-Kutta method or the Adams-Bashforth method. This is because the Euler method is a first-order method, while these other methods are higher-order and have a faster convergence rate. However, the Euler method is often simpler to implement and can still provide accurate approximations in certain cases.

What are the practical applications of understanding the rate of convergence of the Euler method?

Understanding the rate of convergence of the Euler method can help in choosing the appropriate numerical method for solving differential equations. It can also help in determining the appropriate step size to use for a given level of accuracy. Additionally, knowledge of the rate of convergence can aid in the analysis and optimization of numerical algorithms in various scientific and engineering fields.

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