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

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SUMMARY

The discussion clarifies the mathematical concept of splitting the sum from 1 to (n-1) into the difference between the sum from 1 to infinity and the sum from n to infinity. Specifically, it confirms that both sums indeed contain (n-1) terms, as demonstrated with the example where n=5. The confusion arises from a miscount of terms, which is resolved by explicitly calculating the sums. The explanation effectively illustrates the relationship between these sums and reinforces the understanding of infinite series in economics.

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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 [itex]a_1+ a_2+ a_3+ a_4[/itex]. That has 4= n-1 terms, not n-2.

The sum from 1 to infinity would be [itex]1+ a_1+ a_2+ a_3+ a_4+ a_ 5+ a_6+ a_7+ \cdot\cdot\cdot+[/itex] while the sum from n to inifinity is [itex]a_5+ a_6+ a_7+ \cdot\cdot\cdot[/itex]. Subtracting the second from the first leaves [itex]a_1+ a_2+ a_3+ a_4[/itex] as claimed.
 
I see! Thank you soooo much!
 

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