Difference between revisions of "PaperMath’s sum"
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PaperMath’s sum was discovered by the aops user PaperMath, as the name implies. | PaperMath’s sum was discovered by the aops user PaperMath, as the name implies. | ||
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| + | ==See also== | ||
| + | *[[Cyclic sum]] | ||
| + | *[[Summation]] | ||
| + | *[[Symmetric sum]] | ||
| + | |||
| + | [[Category:Algebra]] | ||
| + | [[Category:Definition]] | ||
Revision as of 11:29, 8 October 2023
Contents
PaperMath’s sum
This is a summation identities for decomposition or reconstruction of summations. PaperMath’s sum states,
Or
For all real values of 
, this equation holds true for all nonnegative values of 
. When 
, this reduces to
Proof
We will first prove a easier variant of PaperMath’s sum,
This is the exact same as
But everything is multiplied by 
.
Notice that this is the exact same as saying
Notice that 
Substituting this into yields
Adding 
on both sides yields
Notice that 
As you can see,
Holds true for any values of . Since this is true, we can divide by on both sides to get
And then multiply both sides 
to get
Or
Which proves PaperMath’s sum
Notes
PaperMath’s sum was discovered by the aops user PaperMath, as the name implies.
