is it right to say something like,

"because the infinite series converges, the infinite sequence also converges"?

thanks!

explanations would be helpful if u wish to put some! =)

- Oct 12th 2009, 03:07 AMAskerNotDoes convergence of an infinite series imply convergence of the infinite sequence?
is it right to say something like,

"because the infinite series converges, the infinite sequence also converges"?

thanks!

explanations would be helpful if u wish to put some! =) - Oct 12th 2009, 03:34 AMtonio

If you mean "because the inf. series SUM{a_n} converges then the inf. sequence {a_n} converges" then yes, you can say that....but it is a huge understatement, since if the series converges then not only the sequence converges but in fact it MUST converge to zero.

The other way around is false: a seq. may converge to zero but the inf. series formed with it can easily diverge.

Tonio - Oct 12th 2009, 03:37 AMPlato
That statement is a bit vague. However here is a theorem.

The series $\displaystyle \sum\limits_{k = 1}^\infty {a_k } $ converges only if $\displaystyle \left( {a_n } \right) \to 0$.

Is that what you mean by the above statement?

This is often called*first test for divergence*.

You see that if $\displaystyle \left( {b_n }\right) \not\to 0$ then $\displaystyle \sum\limits_{k = 1}^\infty {b_k } $ must diverge.

On the other hand be careful.$\displaystyle \left( {\frac{1}{n}} \right) \to 0$ BUT $\displaystyle \sum\limits_{k = 1}^\infty {\frac{1}{k}} $**diverges**. - Oct 12th 2009, 04:04 AMtonio
- Oct 12th 2009, 04:20 AMPlato
Tonio, How much do you know about implications?

It is well known that “P implies Q” is logically equivalent to “P only if Q”

These two statements are logically equivalent.

The series $\displaystyle \sum\limits_{k = 1}^\infty {a_k } $ converges only if $\displaystyle \left( {a_n } \right) \to 0$.

If the series $\displaystyle \sum\limits_{k = 1}^\infty {a_k } $ converges then $\displaystyle \left( {a_n } \right) \to 0$ - Oct 12th 2009, 05:06 AMtonio
- Oct 12th 2009, 06:47 AMPlato
If you access to any elementary logic textbook, in English, you will find a discussion of this topic.

There is nice page and a half discussion in__Symbolic Logic__by Irving M Copi

It is also discussed in a Set Theory Text by Stoll (ch. 4)

You can find it in almost any Discrete Mathematics textbook, say; Goodaire, Johnsonbaugh, Ross/Wright, or Grimaldi. - Oct 12th 2009, 10:10 AMtonio

Yes, you were right. It appears in Suppes' "Introduction to logic", where it is explicitily stated that P --> Q can be expressed as "P only if Q".

There's also a nice discussion in No. 10 at Peter Suber, "Translation Tips" which seems to explain my confussion with spoken language. I also calles the "only if" the least intuitive (logical connectives translation) because, apparently, of the language bias we have.

And this is a nice a day since I learned something new.

Thanx

Tonio