Help with serie of function

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In summary, the conversation discusses the convergence of the series \sum_{n=1}^{\infty}\frac{1}{1+n^2x} and its uniform convergence over different intervals. The conversation also mentions the use of the integral convergence criterion for numerical series and the Weierstrass criterion in determining the convergence of the series. The conversation also includes the attempts to find the form of the nth term of the partial sum, but the pattern is not identified.
  • #1
quasar987
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Hi, here the question I have to answer.

Consider the serie of function

[tex]\sum_{n=1}^{\infty}\frac{1}{1+n^2x}[/tex]

a) For which values of x does the serie converges pointwise?

b) Over which intervals of the form [a,b] does the serie converges uniformally? Are there cases where we can have a uniform convergence for a=-infty or b = +infty ?


I answered to a) using the integral convergence criterion for numerical series, but I'm not sure we're allowed to use it since we haven't seen it in class. Is there another way to answer a) ? I first tried to find the form of the nth term of the sequence of partial sums, but I gave up quickly, being unable to see the patern in the sequences {0, 5, 28, 90, ...}, {0, 0, 49, 546,...}, {0, 0, 0, 820, ...}, etc. representing the coefficient of x, x², x^3, etc. respectively.


For b), I found that for x in (0,infty), the serie converges uniformly. I did so by comparing (through the Weierstrass criterion) the serie with the serie of 1/n², which converges. I don't see how to go about testing the other intervals of pointwise convergence for uniform convergence.

Thanks for your time and help.
 
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  • #2
If I try to find the form of the nth term of the partial sum, I get

[tex]S_1(x) = \frac{1}{1+x}[/tex]

[tex]S_2(x) = \frac{2+5x}{(1+x)(1+4x)}[/tex]

[tex]S_3(x) = \frac{3+28x+49x^2}{(1+x)(1+4x)(1+9x)}[/tex]

[tex]S_4(x) = \frac{4+90x+546x^2+820x^3}{(1+x)(1+4x)(1+9x)(1+16x)}[/tex]

...

[tex]S_n(x) = \frac{n+}{(1+x)(1+4x)(1+9x)(1+16x)...(1+n^2x)}[/tex]

Does anyone sees the patern here? :confused:
 
  • #3


Hi there,

Thank you for reaching out for help with your series of functions. It seems like you have made some good progress in answering the questions, so I will try to provide some additional guidance and clarification.

a) To answer this question without using the integral convergence criterion, you can use the definition of pointwise convergence. In order for the series to converge pointwise, the limit as n approaches infinity of the nth term must be equal to 0. In this case, the nth term is 1/(1+n^2x), so we can set this equal to 0 and solve for x. This will give us the values of x for which the series converges pointwise. Keep in mind that this does not guarantee that the series converges, it just tells us where it converges pointwise.

b) To determine the intervals of uniform convergence, you can use the Weierstrass M-test as you have done. However, keep in mind that the M-test only guarantees uniform convergence on intervals of the form [a, b] where a and b are finite. This means that we cannot have uniform convergence for a = -infinity or b = +infinity. To determine the other intervals of uniform convergence, you can use the Cauchy criterion for uniform convergence. This states that a series of functions converges uniformly if and only if for any epsilon > 0, there exists an N such that for all x and n > N, |Sn(x) - S(x)| < epsilon, where Sn(x) is the nth partial sum and S(x) is the sum of the series. This can be a bit more difficult to use, but it can help determine the intervals of uniform convergence in cases where the M-test is inconclusive.

I hope this helps and clarifies some things for you. Keep up the good work and don't hesitate to ask for help if you need it. Good luck with your studies!
 

1. What is a series of functions?

A series of functions is a sequence of functions where the output of one function becomes the input of the next function in the sequence. This allows for complex calculations and problem-solving to be broken down into smaller, more manageable steps.

2. How do I determine the domain and range of a series of functions?

The domain of a series of functions is the set of all possible input values that can be used in the sequence. The range is the set of all possible output values that can be produced by the sequence. To determine the domain and range, you must consider the restrictions and limitations of each individual function in the series.

3. Can a series of functions have different types of functions in it?

Yes, a series of functions can include a variety of functions such as linear, quadratic, exponential, trigonometric, and more. The type of function used in the series depends on the problem being solved and the desired outcome.

4. How can I simplify a series of functions?

To simplify a series of functions, you can use mathematical techniques such as factoring, combining like terms, and applying the rules of exponents. You can also consider using a graphing calculator or software to help visualize and simplify the functions.

5. What are some common applications of series of functions?

Series of functions are commonly used in physics, engineering, economics, and other fields to model and solve complex problems. They are also used in computer programming and data analysis to process and manipulate large sets of data.

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