Infinite Sum

  • Thread starter fderingoz
  • Start date
  • #1
13
0

Homework Statement



Is the proposition

[itex]\sum^{\infty}_{n=1}[/itex]|x[itex]_{n}[/itex]|=0 ⇔[itex]\forall[/itex]n[itex]\in[/itex]IN x[itex]_{n}[/itex]=0

true? If it is true how can we prove that ?

Homework Equations





The Attempt at a Solution


I proved the [itex]\Leftarrow[/itex] side of proposition but i could not prove the [itex]\Rightarrow[/itex] side of proposition.
 

Answers and Replies

  • #2
938
9
Is it not true that [itex] \sum_n |x_n| > x_m [/itex] for all m?
 
  • #3
35,135
6,884

Homework Statement



Is the proposition

[itex]\sum^{\infty}_{n=1}[/itex]|x[itex]_{n}[/itex]|=0 ⇔[itex]\forall[/itex]n[itex]\in[/itex]IN x[itex]_{n}[/itex]=0

true?
Is this what you're trying to prove?
$$ \sum^{\infty}_{n=1}|x_{n}|= 0 \iff \forall n \in Z,~ x_{n} = 0$$

I wasn't sure what you meant by IN. Also, one symbol you used (⇔) renders as a box in my browser.
Edit: Now it's showing up. That's odd, it didn't before.

Tip: Use one pair of tex or itex tags for the whole expression, rather than a whole bunch of them.
If it is true how can we prove that ?

Homework Equations





The Attempt at a Solution


I proved the [itex]\Leftarrow[/itex] side of proposition but i could not prove the [itex]\Rightarrow[/itex] side of proposition.
 
Last edited:
  • #4
13
0
Mark44

The proposition which i want to prove is exactly the proposition which you write. I meant natural numbers by IN. Thanks for your suggestions.

I am waiting for your answers.
 
  • #5
Dick
Science Advisor
Homework Helper
26,263
619
Mark44

The proposition which i want to prove is exactly the proposition which you write. I meant natural numbers by IN. Thanks for your suggestions.

I am waiting for your answers.

Try a proof by contradiction. Assume one of the a's isn't zero.
 
  • #6
13
0
[itex]\sum\limits_{n=1}^{\infty}|x_{n}|=0\Rightarrow\lim \limits_{n\rightarrow\infty}\sum\limits_{k=1}^{n}|x_{k}|=0[/itex]. Since [itex]s_{n}= \sum\limits_{k=1}^{n}|x_{k}| [/itex] is an increasing sequence it can converge only its supremum. So [itex] \sup\limits_{n}s_{n}=0 [/itex]. Thus [itex] \forall n,\, 0\leq s_{n} = \sum\limits_{k=1}^{n}|x_{k}|\leq 0 [/itex] which means [itex] \forall n,\, x_{n}=0 [/itex].

I made this proof. What do you think is there any mistake in this proof ?
 

Related Threads on Infinite Sum

  • Last Post
Replies
1
Views
749
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
4
Views
3K
  • Last Post
Replies
4
Views
524
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
0
Views
1K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
6
Views
801
  • Last Post
Replies
2
Views
871
  • Last Post
Replies
13
Views
1K
Top