Examples of u_n and v_n Sequences with Finite Sum | Help with Homework"

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In summary, the conversation discusses the search for examples of u_n and v_n that satisfy the condition u_{n+1}\leq u_n+v_n for all n and have a finite sum for v_n. The conversation also mentions a theorem stating that if the sum of v_n is finite, then lim_{n\rightarrow \infty}u_n exists. A suggestion is made to pick a nice, summable v_n and a decreasing u_n to guarantee the validity of the theorem.
  • #1
mercedesbenz
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Homework Statement


Please help me find examples.

[tex]u_n\subset [0,\infty)[/tex] and [tex]v_n\subset [0,\infty)[/tex] such that
[tex]u_{n+1}\leq u_n+v_n[/tex] for all n and [tex]\sum_{n=1}^{\infty}v_n[/tex]
is finite.


Homework Equations





The Attempt at a Solution

 
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  • #2
Say you take u_n=0 for all n...
 
  • #3
morphism said:
Say you take u_n=0 for all n...

I want to find [tex]v_n\neq 0 ,u_n\neq 0[/tex] for all n
 
  • #4
My advice is to first pick a nice, summable v_n.
 
  • #5
Please help me find examples.

[tex]u_n\subset [0,\infty)[/tex] and [tex]v_n\subset [0,\infty)[/tex] such that
[tex]u_{n+1}\leq u_n+v_n[/tex] for all n and [tex]\sum_{n=1}^{\infty}v_n[/tex]
is finite.

In fact, this is a theorem which say that

[tex]u_n\subset [0,\infty)[/tex] and [tex]v_n\subset [0,\infty)[/tex] such that
[tex]u_{n+1}\leq u_n+v_n[/tex] for all n
If [tex]\sum_{n=1}^{\infty}v_n[/tex]
is finite then [tex]\displaystyle{\lim_{n\rightarrow \infty}u_n}[/tex] exists

then I want to find [tex]u_n[/tex] which dificult to find lim in order to guarantee
this therem is well better than MCT because this therem is generalization of MCT.
 
  • #6
Why don't you want to put in some effort?

Pick a summable (v_n), like say v_n = 1/2^n. Now pick any decreasing (u_n), like u_n = 1/n.
 

1. What is the definition of a sequence with finite sum?

A sequence with finite sum is a sequence in which the terms eventually decrease and approach a specific value, resulting in a sum that is finite or finite in value. This means that the sequence has a finite or fixed number of terms, and the sum of those terms is a finite number.

2. Can you provide an example of a un sequence with finite sum?

Yes, an example of a un sequence with finite sum is the sequence 1, 1/2, 1/4, 1/8, 1/16, ... Each term in this sequence is half the value of the previous term, resulting in a finite sum of 2.

3. Can you provide an example of a vn sequence with finite sum?

Yes, an example of a vn sequence with finite sum is the sequence 2, 4, 6, 8, 10, ... Each term in this sequence is 2 more than the previous term, resulting in a finite sum of 100.

4. How can examples of un and vn sequences with finite sum be used to help with homework?

Examples of un and vn sequences with finite sum can be used to better understand the concept of sequences with finite sum and to practice solving problems involving such sequences. These examples can also serve as a guide for creating your own sequences with finite sum for homework assignments.

5. Are there any common mistakes to avoid when working with un and vn sequences with finite sum?

Yes, some common mistakes to avoid when working with un and vn sequences with finite sum include not correctly identifying the pattern or rule of the sequence, not considering the specified number of terms in the sequence, and not paying attention to the signs of the terms (positive or negative). It is important to carefully analyze the problem and the sequence to avoid these mistakes.

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