0

I am trying to evaluate the following summation:

Please look at page 3 in the book link

Adventures in stochastic Processes, page 3

When he is deriving the expectation of a geometric random variable he writes E(X) as a double summation and then changes the order of summation. My doubt is when can you and when can you not reverse the order of summations and HOW DO YOU DETERMINE the limits of summation when you change the order of summation?

I did not understand how the limits of the indices j and k change when you reverse the order of summations..please help

flag
I'm tempted to recommend a look in Rudin's Principles of Mathematical Analysis, but: 1) I don't have a copy to hand, so can't say for sure that it would help; 2) I don't know if you have easy access to a copy; 3) I don't know from your question what level of studies you're at, so don't know how helpful you'd find such a reference. – Yemon Choi Jul 13 2010 at 7:04
1 
 Also, your question might fit better on a site like NRICH or one of the others mentioned in the FAQ (mathoverflow.net/faq#whatnot ). Note in particular that MO is not at present meant to be a site for any and all mathematical questions (at least as I understand things). – Yemon Choi Jul 13 2010 at 7:06
2 
 To give you some help at least, the rule of thumb is that interchanges of summation are fine if you are working with absolutely convergent series. As for how the ranges of summation are changed, if you're still stuck then I suggest you try one of the sites I suggested – Yemon Choi Jul 13 2010 at 7:07
2 
 You'd profit most from working this one out yourself before looking it up. Think about summing over the coordinates (j,k) in the plane. In what ways can you do this? (As already stated, it's a good question for a site like NRICH.) – Derek Jennings Jul 13 2010 at 7:21
1 
 The question has been closed. It is not really appropriate for our site, but you will probably get help if you ask it at NRICH or one of the other sites listed in the FAQ. – Pete L. Clark Jul 13 2010 at 11:20

closed as too localized by Yemon Choi, Wadim Zudilin, Gjergji Zaimi, Franz Lemmermeyer, Pete L. Clark Jul 13 2010 at 11:19

2 Answers

0

All the terms in this summation are positive, so you can change the order. The result wouldn't change.

The summation is done over all pairs $(k, j)$ with $j \le k$. The first summation first sums over all $k$ and after that over all $j$, the second one --- first sums over all $j$ and after that over all $k$.

link|flag
3 
 I think there is a vague consensus policy on MO that one should avoid leaving answers to questions which are "too basic", in case it encourages people to keep asking similar questions. However, this is only my personal take and others may disagree. – Yemon Choi Jul 13 2010 at 8:52
I think it is unfair to downvote correct answers just because you consider them "too easy"! This is why some people think mathematicians are an arrogant bunch - think about the impact of your behaviour! – vonjd Jul 13 2010 at 8:58
4 
 @vonjd: I have not downvoted this answer, so I am not sure why you've made this comment. This is also not about arrogance but about trying to keep the site on track. I can only apologize for not being an interested amateur. – Yemon Choi Jul 13 2010 at 9:02
4 
 I did that, not Yemon. Does this answer really improve on the comments made above? – Franz Lemmermeyer Jul 13 2010 at 9:03
3 
 Please do not answer questions that instead should be closed! – Andres Caicedo Jul 13 2010 at 14:08
show 3 more comments
0

This is because of the commutativity property. Have a look here under "double infinite summation": http://functions.wolfram.com/GeneralIdentities/12/

link|flag
3 
 That does it. Voting to close. – Franz Lemmermeyer Jul 13 2010 at 9:11

Not the answer you're looking for? Browse other questions tagged or ask your own question.