Solve Rate of Convergence Problems Easily

In summary, the conversation discusses finding rates of convergence and solving problems involving trig functions. The book suggests using rates in the form 1/n^p, while the speaker also mentions using the Maclaurin polynomial. They also mention the formula \lim_{x\rightarrow 0}\frac{sin(x)}{x} = 1 and its relation to the rate of convergence. They also express difficulty in solving \lim_{n\rightarrow infinity} [ln(n+1) - ln(n)] = 0 and ask for help and resources on this topic.
  • #1
Zaphodx57x
31
0
I'm not sure how to solve these problems. The example given in the book does not use trig functions. Any insight into how I solve these would be helpful.

Find the following rates of convergence.
[tex]
\lim_{n\rightarrow infinity} sin(1/n) = 0
[/tex]
My thought would be to do the following
[tex]
|sin(1/n) - 0| <= 1
[/tex]
But the book says to get a rate in the form [tex]1/n^p[/tex]

The following also gives me trouble.
[tex]
\lim_{n\rightarrow infinity} sin(1/n^2) = 0
[/tex]
which seems like it should converge faster than the the first one.
 
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  • #2
I made some progress by taking the maclaurin polynomial and only keeping the first couple terms. I can't get anything satisfactory for this one though
[tex]\lim_{n\rightarrow infinity} [ln(n+1) - ln(n)] = 0[/tex]
I get to an answer of 2-n or so, maybe I should keep more terms.
Anybody help would be appreciated.
 
  • #3
Do you know that [tex]\lim_{x\rightarrow 0}\frac{sin(x)}{x} = 1[/tex]? If you let x= 1/n, that's the same as [tex]\lim_{n\rightarrow \infty}\frac{sin(1/n)}{1/n}= 1[/tex]. What does that tell you about the rate of convergence?

To do sin(1/n2), look at [tex]\frac{sin(1/n^2}{1/n^2}[/tex]
 
  • #4
im searching for tutorials on this section particularly...
any links?
 

What is the rate of convergence?

The rate of convergence is a measure that indicates how quickly a sequence of numbers or values approaches a specific limit or target value. It is typically expressed as a decimal or a percentage.

Why is it important to calculate the rate of convergence?

Calculating the rate of convergence allows us to determine how quickly an algorithm or method is converging towards a solution. This can help us evaluate the efficiency and accuracy of different approaches and make informed decisions about which method to use.

What factors can affect the rate of convergence?

The rate of convergence can be affected by various factors such as the initial guess, the choice of algorithm or method, the complexity of the problem, and the precision of the calculations. In some cases, the rate of convergence may also depend on the properties of the function or system being solved.

How can we improve the rate of convergence?

One way to improve the rate of convergence is to choose a more efficient algorithm or method for solving the problem. This could involve using a different approach or adjusting the parameters of the algorithm. Additionally, improving the precision of the calculations or choosing a better initial guess can also help improve the rate of convergence.

Are there any limitations to calculating the rate of convergence?

Yes, there are some limitations to calculating the rate of convergence. In some cases, the rate of convergence may not be well-defined or may be too difficult to calculate. Additionally, the rate of convergence may not accurately reflect the performance of an algorithm or method in all cases, as it may depend on specific conditions or assumptions.

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