Double summation order changed

[Mahavishnu]

Hi, can someone please clarify why is it ok to change the order of the sum from k = 0 to infinity, of the sum from t = 0 to infinity for the sum from t = 0 to infinity, of the sum from k = t to infinity?

I hope I made myself clear. Thanks in advance!

 

[stapel]

Hi, can someone please clarify why is it ok to change the order of the sum from k = 0 to infinity, of the sum from t = 0 to infinity for the sum from t = 0 to infinity, of the sum from k = t to infinity?

I hope I made myself clear. Thanks in advance!

It might help if we knew what the summation was, etc...? Thank you!

 

[Mahavishnu]

Sure, I'm sorry. I've attached a picture of the textbook where it comes from. It is the formula for an annuity subject to survival, where v^t is the discount factor and k|qx is the probability that an individual of age x, survives to age x+k and dies before x+k+1.

If you need more information, let me know.

Thank you!

 

[mmm4444bot]

Your image is blurry. (This forum reduces over-sized images.) We get the best results by first cropping off all the irrelevant parts.

 

[Mahavishnu]

I'm sorry, I just noticed because I was replying from my phone.

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\(\displaystyle \large{ \displaystyle \sum_{k=0}^{\infty}\, \sum_{t=0}^k\, v^t\; {}_k|q_x\, =\, \sum_{t=0}^{\infty}\, \sum_{k=t}^{\infty}\, v^t\; {}_k|q_x }\)

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Is it better?

Anyway, the left hand side is the summation from k=0 to infinity of the summation from t = 0 to k of v^t times k|qx, and the right hand side is the summation from t=0 to infinity of the summation from k = t to infinity of v^t times k|qx. So it is the same, just changing the order of sums.

Thank you!

 

[mmm4444bot]

Is it better?

You can't tell? :???:

 

[Dr.Peterson]

\(\displaystyle \large{ \displaystyle \sum_{k=0}^{\infty}\, \sum_{t=0}^k\, v^t\; {}_k|q_x\, =\, \sum_{t=0}^{\infty}\, \sum_{k=t}^{\infty}\, v^t\; {}_k|q_x }\)

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...the left hand side is the summation from k=0 to infinity of the summation from t = 0 to k of v^t times k|qx, and the right hand side is the summation from t=0 to infinity of the summation from k = t to infinity of v^t times k|qx. So it is the same, just changing the order of sums.

As I had suspected, you originally described the sums incorrectly, saying that all were "to infinity". As it is, do you notice that the original summation covers points (k,t) in an octant of the plane (an infinite "triangle"), and the rewritten summation covers the same region? All they are doing is rearranging the order of summation, which is valid as long as it converges.

 

[Mahavishnu]

As I had suspected, you originally described the sums incorrectly, saying that all were "to infinity". As it is, do you notice that the original summation covers points (k,t) in an octant of the plane (an infinite "triangle"), and the rewritten summation covers the same region? All they are doing is rearranging the order of summation, which is valid as long as it converges.

Thanks, you've been really helpful!