Proving Theorem: "If a Sequence Converts...

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In summary, the conversation discusses a theorem and a proof involving sequences and summation. The proof uses a formula called "summation by parts" and the conversation ends with a request for further explanation on the thought process behind using this formula.
  • #1
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Homework Statement



I recently posted a question asking to prove the following theorem:

"If a sequence a_1 + a_2 + ... converges and if b_1, b_2, ... is a bounded monotonic sequence of numbers, then (a_1)(b_1) + (a_2)(b_2) + ... converges"

Here is a proof that I came across for it:

Let s_n denote the partial sums of [tex] \sum_{v=1}^n a_v [/tex], s the sum, and let [tex] \xi_n = s_n - s [/tex]. Then [tex] \sum_{v=n}^m a_v b_v = \sum_{v=n}^m (\xi_v - \xi_{v-1}) b_v = \sum_{v=n}^m \xi_v(b_v - b_{v+1}) - \xi_{n-1} b_n + \xi_m b_{m+1} [/tex].

For every sufficiently large v, [tex] |\xi_v| < \epsilon [/tex], and

[tex] \sum_{v=n}^m a_v b_v < \epsilon \sum_{v=n}^m |b_v - b_{v+1}| + \epsilon |b_n| + \epsilon |b_{m+1}| < \epsilon |b_n - b_{m+1}| + \epsilon |b_n| + \epsilon |b_{m+1}| [/tex].

This is in turn less than [tex]4B \epsilon [/tex], where B is a bound for |b_v|, and the series [tex] \sum_{v=1}^{\\infty} a_v b_v [/tex] converges


Homework Equations





The Attempt at a Solution



I understand the proof and everything. I was wondering though, how did the writer of the proof know to rewrite the sum as this: [tex] \sum_{v=n}^m a_v b_v = \sum_{v=n}^m (\xi_v - \xi_{v-1}) b_v = \sum_{v=n}^m \xi_v(b_v - b_{v+1}) - \xi_{n-1} b_n + \xi_m b_{m+1} [/tex] ?

It just seems so random, something that I never would've thought about. If you could, could you please explain the thought processes he went through to realize he had to rewrite the sum in that form?

Thanks
 
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  • #2
That formula is called "summation by parts." It is the discrete analog of integration by parts.
 
  • #3
Ah, thanks! The wiki article on it isn't loading properly for whatever reason. I'll wait until tomorrow to give it another look. If I can't find what I'm looking for on the formula, I'll be back!

Thanks
 

1. What is a theorem?

A theorem is a statement that has been proven to be true using logic and mathematical reasoning.

2. What does it mean for a sequence to convert?

A sequence is said to convert if its terms approach a specific value as the number of terms increases.

3. How do you prove the theorem "If a sequence converts, then its limit is unique"?

This theorem can be proven using the definition of convergence of a sequence, along with the triangle inequality and the properties of limits.

4. Can the converse of this theorem also be proven?

Yes, the converse of this theorem can also be proven. It states that if a sequence has a unique limit, then it must converge.

5. What are some real-world applications of this theorem?

This theorem has applications in fields such as physics, engineering, and finance, where it is used to analyze and predict the behavior of various systems and phenomena.

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