Help with these complex analysis series problems?

In summary: Thank you again for your help!In summary, the problem involves showing that rearrangements of a convergent series can converge to any value and that there are three possible scenarios for the set of values that the rearranged series can converge to. Part B requires finding the sum of a convergent series by grouping terms in a specific way, while part C involves expressing the rearranged series in a specific form and finding its infinite sum. Finally, part D asks to show that for any convergent alternating series with real values, the sum can be rearranged to converge to an arbitrary limit. These problems may be challenging, but with perseverance and problem-solving skills, they can be solved.
  • #1
nontradstuden
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I don't even know where to start or go with this first problem.

A) Assume that 'a sub n' E C and consider rearrangements of the convergent series the 'sum of 'a sub n' from n=1 to infinity'. Show that each of the following situations is possible and that this list includes all possibilities.

1) The sum of a sub n from n=1 to infinity converages absolutely and hence all rearrangements converge to the same value.

2) The set of possible values of convergent rearrangements is all of C.

3) The set of possible values of convergent rearrangements is an arbitrary line in C.

B) Show that the sum of [ (-1)^(n+1) / n ] from n=1 to infinity converges conditionally. The value of the sum is log(2), approcimately .69. Show that by grouping the terms by taking two positive terms, then one negative term, then two positive terms, then one negative term, and so on, the series adds up to a number larger than 1. The value of the sum therefore depends on the order in which the terms are summed.

This is all that I could come up with:

The abs value of the series diverges because it's the harmonic series where p=1.
The series itself converges by the alternating series test because the nth term converges to zero and the series is decreasing. Therefore the given series converges conditionally.

Now for the rearranging:

1 + 1/3 -1/2 + 1/5 + 1/7 -1/4 and so forth. I then just listed the partial sums.

s1= 1 + 1/3 -1/2= 5/6
s2= s1 + 1/5 + 1/7 -1/4= 0.926
s3= s2 + 1/13 + 1/15 - 1/8= 0.98013
and so on...

I know that this isn't right. I don't know how I'm supposed to show that the sum of the rearranged series is greater than 1.

C) Express the rearranged series *found above in B* in the form 'the sum of b sub n from n=1 to infinity', and find a simple expression for b sub n. Try to find the infinite sum.

After a lot of erasing I came up with:


1 + 1/3 -1/2 + 1/5 + 1/7 -1/4 and so forth can be expressed as

the sum of 1/ (4k-3) + 1/(4k-1) - 1/(2k) from k=1 to infinity. I don't know if this is correct for 'b sub n' form or how to find the infinite sum of this rearranged series.

D) Assume that 'a sub n' E R(real) and that the sum of 'a sub n' from n=1 to infinity converged conditionally. Fix an arbitrary L(limit) E R(real). Show that we can rearrange the terms to make the sum converge to L.
I don't know where to start or where to go with this one, too. Do I treat 'a sub n' as an alternating series that converges conditionally?

I know that you aren't here to solve it for me, but any help you can give me would be great. Thank you very much.

Please, let me know if I need to clarify anything.
 
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  • #2
For part A, unless you were given a specific sequence an to look at, the problem is asking you to use the definition of a convergent series to show that any convergent series can be rearranged to converge to any value you want. I think this is similar to the http://mathworld.wolfram.com/RiemannSeriesTheorem.html" [Broken].

For part B, you are on the right track. You'll need to calculate out to s7 to show it is > 1.

For C, try turning your new formula into a single term and see if you can sum it.

The way I think about D is that, for any convergent alternating series, we have an infinite number of positive terms and an infinite number of negative terms. Consider the alternating harmonic series. Let's say we want to make it equal to 100. We can take all of the positive (odd) terms that we want, and add them together to get to eventually reach 100. Then we add -1/2. Then we add more positive terms to get back to 100. Then we add -1/4. Then we add more positive terms to get back to 100, and so on. Since we never run out of positive terms, we can keep going with this insane ratio of positive terms to negative terms forever. Hence, the series converges to any limit that we want.
 
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  • #3
@Kru.

Thanks!

Your response helped a lot.
I will rework the problems. This is my first analysis course, so it is tough for me.


Edit:

I just wanted to say that I asked my professor about these problems today and he said that he didn't know they were this difficult.

He said that 'B' was more difficult/ time consuming than he originally thought. I feel good about the little problem solving that I did do. It took a while just for me to come up with that formula. I am going to finish it up now.
 
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1. What is complex analysis?

Complex analysis is a branch of mathematics that deals with the study of complex numbers and functions. It is a powerful tool used to analyze functions that are defined over the complex plane.

2. What are some common applications of complex analysis?

Complex analysis has many practical applications in fields such as engineering, physics, and economics. It is used to solve problems involving electric circuits, fluid dynamics, and signal processing, among others.

3. How do I approach solving complex analysis series problems?

First, make sure you have a strong understanding of complex numbers, functions, and series. Then, carefully read the problem and identify any patterns or relationships. Utilize various techniques such as Cauchy's integral theorem or the Cauchy-Riemann equations to solve the problem step-by-step.

4. What are some common challenges when working with complex analysis series problems?

Some common challenges include understanding the properties of complex numbers and functions, determining which techniques to use, and keeping track of complex calculations. It is important to have a strong foundation in complex analysis and to practice regularly to overcome these challenges.

5. Are there any resources available for help with complex analysis series problems?

Yes, there are many resources available such as textbooks, online tutorials, and practice problems. You can also seek assistance from your professor or classmates, or attend study groups or tutoring sessions. It is important to actively seek help and practice regularly to improve your skills in solving complex analysis series problems.

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