Solve Convergence of Series Problem: Find Bounds for x_0

• quasar987
In summary, the conversation is about analyzing the convergence of a series of functions and determining whether it converges pointwise and uniformly in a given interval. The speaker suggests considering a given number and using the Weirstrass M-test to determine uniform convergence. However, there are singularities in the interval that prevent uniform convergence. The speaker also mentions finding a way to prove pointwise convergence and the potential dependence on x near a singular value. The conversation ends with a request for a quick answer as the question needs to be handed out the next day.
quasar987
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I need help with the following problem.

Consider the serie of function

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

The serie is undefined for $x \in \{0\}\cup \{-1/n^2, \ n\in \mathbb{N}\}$. I want to find wheter it converges pointwise in (-1, 0) or not and if it does, does it converge uniformly?

The way I would start this problem is by saying: For a given number $m \in \mathbb{N}$, consider

$$x_0 \in \left(\frac{-1}{m^2} \ ,\frac{-1}{(m+1)^2}\right)$$

Consider

$$f_n(x) = \frac{1}{1+n^2x}$$

Then

$$|f_n(x_0)| = \frac{1}{|1+n^2x_0|} = \frac{1}{|1-n^2|x_0||}= \left\{ \begin{array}{rcl} \frac{1}{1-n^2|x_0|} & \mbox{for} & n<\sqrt{\frac{1}{|x_0|} \\ \frac{1}{n^2|x_0|-1} & \mbox{for} & n>\sqrt{\frac{1}{|x_0|} \end{array}\right$$

and

$$\sum_{n=1}^{\infty}|f_n(x_0)| = \sum_{n=1}^{\left[\sqrt{1/|x_0|}\right]}\frac{1}{1-n^2|x_0|} + \sum_{n=\left[\sqrt{1/|x_0|}\right]+1}^{\infty}\frac{1}{n^2|x_0|-1}$$

I'm guessing this serie converges, but I'm having trouble finding a convergent serie to bound it with. The other convergence tests have failed and the use of the integral convergence criterion is forbiden. I know that if there is a serie to bound it with, it would be of the form

$$\sum_{n=1}^{\infty}a_n = \sum_{n=1}^{\left[\sqrt{1/|x_0|}\right]}\frac{1}{1-n^2|x_0|} + \sum_{n=\left[\sqrt{1/|x_0|}\right]+1}^{\infty}b_n$$

with

$$\frac{1}{n^2|x_0|-1} \leq b_n$$

for n > N.

Edit:

And if there exists such an N that also satisfies

$$N\leq \left[\sqrt{1/|x_0|}\right]$$

then according to Weirstrass M-test, the convergence is uniform.

Last edited:
Because of all the singularities (0, -1/n2) in the interval, it can't converge uniformly.

quasar987 said:
The serie is undefined for $x \in \{0\}\cup \{-1/n^2, \ n\in \mathbb{N}\}$. I want to find wheter it converges pointwise in (-1, 0) or not and if it does, does it converge uniformly?

That was not well said. What I meant to say is, does it converge pointwise and uniformly for the intervals in (-1,0) where the serie is defined. I.e. in the intervals

$$\left(\frac{-1}{m^2} \ ,\frac{-1}{(m+1)^2}\right), & m \in \mathbb{N}$$

In the intervals of interest it converges pointwise, but not uniformly because of the blow ups at the end points of each interval.

On the basis of which theorem(s) are these statements made true? I would apreciate a quick answer because I need to hand out this question tomorrow!

Thanks!

Last edited:
By the way, I have found how to prove the pointwise convergence, I just don't know how to prove that it's not uniformly convergent on these intervals.

I don't know what approach you are using to prove pointwise convergence. However, if you are using the old fashioned epsilon delta argument, you will see that there is a dependence on x when x is near a singular value.

I noticed that like 10 minutes before handing it out

1. What is the definition of convergence for a series?

The convergence of a series is the property that the sum of its terms approaches a finite limit as the number of terms increases.

2. How do you determine if a series is convergent or divergent?

A series is convergent if its sum approaches a finite limit as the number of terms increases. To determine if a series is convergent or divergent, we can use various tests such as the comparison test, ratio test, or integral test.

3. What is the role of x_0 in finding the bounds of convergence for a series?

x_0 represents the starting point of the series and is used to determine the range of values for which the series will converge. It is important to find the bounds of convergence to ensure that the series will converge for all values within that range.

4. How do you find the bounds of convergence for a series?

To find the bounds of convergence for a series, we can use the ratio test or the root test. These tests involve taking the limit of the absolute value of the ratio or root of consecutive terms in the series. If the limit is less than one, the series will converge. Otherwise, it will diverge.

5. Can a series have more than one bound of convergence?

Yes, a series can have multiple bounds of convergence. This means that the series will converge for all values between these bounds and will diverge for values outside of the range. It is important to find all bounds of convergence to accurately determine the convergence of the series.

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