Why does splitting 1 to (n-1) into (1 to infinity) minus (n to infinity) work?

[toni]

I got this from my Economics lecture notes...

How come 1 to (n-1) can be split up into [(1 to infinity) minus (n to infinity)]?

What confuses me is the former one has (n-2) terms, but the latter one has only (n-1) terms...

And it doesn't make sense to me graphically...

 

[HallsofIvy]

The thing that is confusing you is not true. Both sums have n-1 terms.

Did you try seeing what happens with specific n? If n= 5, the sum from 1 to n-1= 4 is $a_1+ a_2+ a_3+ a_4$. That has 4= n-1 terms, not n-2.

The sum from 1 to infinity would be $1+ a_1+ a_2+ a_3+ a_4+ a_ 5+ a_6+ a_7+ \cdot\cdot\cdot+$ while the sum from n to inifinity is $a_5+ a_6+ a_7+ \cdot\cdot\cdot$. Subtracting the second from the first leaves $a_1+ a_2+ a_3+ a_4$ as claimed.

 

[toni]

I see! Thank you soooo much!