Does the Series Converge Pointwise and Uniformly for Certain Values of x?

[quasar987]

Hi, here the question I have to answer.

Consider the serie of function

$$\sum_{n=1}^{\infty}\frac{1}{1+n^2x}$$

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.

 

[quasar987]

If I try to find the form of the nth term of the partial sum, I get

$$S_1(x) = \frac{1}{1+x}$$

$$S_2(x) = \frac{2+5x}{(1+x)(1+4x)}$$

$$S_3(x) = \frac{3+28x+49x^2}{(1+x)(1+4x)(1+9x)}$$

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

...

$$S_n(x) = \frac{n+}{(1+x)(1+4x)(1+9x)(1+16x)...(1+n^2x)}$$

Does anyone sees the patern here?

 

[thepatientmental]

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!