Math Is Fun Forum

Dragonshade

Member

Registered: 2008-01-16

Posts: 147

A quick question about series

You have a series

I read it on the book, says, if you find the expression for partial sum for the series then take the limit, if it be the sum of the series

And I also know rearranging terms is not valid method of dealing with series

but what if I rearrange the term to get the expression for partial sum, and then take the limit, it sounds valid, but I am not sure

like rearranging that to

Last edited by Dragonshade (2008-11-11 19:45:21)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: A quick question about series

And I also know rearranging terms is not valid method of dealing with series

It depends. If your series is absolutely convergent, you can rearrange the terms. If it is only conditionally convergent, then you cannot rearrange the terms.

Last edited by JaneFairfax (2008-11-12 01:32:48)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: A quick question about series

If it is only conditionally convergent, then you cannot rearrange the terms.

You are not guaranteed to be allowed to rearrange terms.  However, some rearrangements are allowed.  In fact, you are allowed to rearrange any number of finitely many terms.  Some infinite rearrangements may be allowed as well, but it depends on the series.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Dragonshade

Member

Registered: 2008-01-16

Posts: 147

Re: A quick question about series

Oh, I see.  Thanks guys

Dragonshade

Member

Registered: 2008-01-16

Posts: 147

Re: A quick question about series

so  to prove

is divergent

It's not valid to split it to

since the latter part diverges

so the only way to do it is

so it diverages

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: A quick question about series

If it is only conditionally convergent, then you cannot rearrange the terms.

You are not guaranteed to be allowed to rearrange terms.  However, some rearrangements are allowed.  In fact, you are allowed to rearrange any number of finitely many terms.  Some infinite rearrangements may be allowed as well, but it depends on the series.

Yes, of course you can rearrange finitely many terms. I should have said that I meant rearranging an infinite number of terms.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: A quick question about series

so  to prove

is divergent

It's not valid to split it to

since the latter part diverges

so the only way to do it is

so it diverages

You got it.