Understanding the Splitting of (1 to (n-1)) in Economics Lecture Notes

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The discussion clarifies the mathematical splitting of the sum from 1 to (n-1) into the expression [(1 to infinity) minus (n to infinity)]. It addresses confusion regarding the number of terms in each sum, confirming that both sums indeed contain (n-1) terms. A specific example with n=5 illustrates that the sum from 1 to 4 has four terms, aligning with the formula. The subtraction of the sum from n to infinity from the sum from 1 to infinity accurately results in the original sum from 1 to (n-1). This explanation resolves the initial confusion about the term count and graphical representation.
toni
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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...
 

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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.
 
I see! Thank you soooo much!
 

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