Diverging and converging infinite series

In summary, the conversation discusses the use of tests for convergence and divergence in a series, specifically the ratio test and limit test. The person is unsure about their method for applying the limit test and asks for clarification. They also mention using a calculator to find the value of a function and forming an equation to approximate the function using Newton's method. They have a question about finding the constant k in the equation.
  • #1
eptheta
65
0
I looked through some tutorials to find intervals of divergence and tests for divergence...
My series:
[PLAIN]http://img843.imageshack.us/img843/4193/51453212.jpg
a and x are constants...
I did the ratio test and i get [tex]\rho[/tex]=1, so i tried to apply the limit test to see if an is zero or does not exist... The problem is, i am not sure about my method...
Do i divide the numerator(1) by n also, if so, the test gives an=0... But then again i am not sure...
I learned about these tests from http://www.math.unh.edu/~jjp/radius/radius.html"...

Could someone please tell me if this sum is convergent or divergent (and the method you used to find this out)

Thanks
 
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  • #3
Take x outside the summation. The terms in the sum behave like 1/n for large n. This series is well known divergent. The sum behaves like ln(n).
 
  • #4
That makes sense, thanks.. I have one more question...

I was fiddling with my calculator and typed "Ans-(Ans-[tex]\sqrt{Ans}[/tex]-2)" and kept clicking =...(The expression is in the same form as xn - f(x)/f'(x) but is just equal to [tex]\sqrt{Ans}[/tex]+2...
I found that it converges as 4. It seemed logical enough...But then i decided to form the equation that represents f(x)...

so basically, what i got out of the expression is that y/(dy/dx)=x-[tex]\sqrt{x}[/tex]-2
dy/y=dx/(x-[tex]\sqrt{x}[/tex]-2)
Integrating(painfully), ln y=k(x-[tex]\sqrt{x}[/tex]-2)([tex]\sqrt{x}[/tex]-2/[tex]\sqrt{x}[/tex]+2)1/3

Its root x -2 in the numerator not root 3 -2...

Where k is a constant... I should have tried with something simpler, but is this the correct way to form an approximation of a function that converges using Newton's method (or am i just being an oaf)
If it is the former, then how do i find k.
If it is the latter, then how do i find f(x) for :
[PLAIN]http://img714.imageshack.us/img714/5351/96407726.png

Thanks.
 
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  • #5
Sorry, it is not summation to infinity, but rather term at infinity for the last expression.
-Thanks
 

Related to Diverging and converging infinite series

What is a diverging infinite series?

A diverging infinite series is a series where the terms do not approach a finite limit as the number of terms approaches infinity. This means that the sum of the series will approach infinity or negative infinity, depending on the behavior of the terms.

What is a converging infinite series?

A converging infinite series is a series where the terms approach a finite limit as the number of terms approaches infinity. This means that the sum of the series will approach a finite value as the number of terms increases.

How do you determine if an infinite series is divergent or convergent?

To determine if an infinite series is convergent or divergent, you can use various tests such as the ratio test, the root test, or the integral test. These tests use the properties of the series' terms to determine if the series will approach a finite value or infinity.

What is the difference between absolute and conditional convergence?

Absolute convergence refers to a series where the sum of the absolute values of the terms converges to a finite value. Conditional convergence refers to a series where the sum of the terms converges, but not the absolute values of the terms. In other words, the order in which the terms are added affects the sum of the series.

Why are diverging and converging infinite series important in mathematics?

Infinite series are used to represent functions and can help us understand the behavior of these functions. Diverging and converging infinite series can help us determine if a function is continuous, differentiable, or integrable. They also have applications in fields such as physics, engineering, and economics.

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