- #1
embphysics
- 67
- 0
Hello,
The problem to be solved is [itex]x^n - y^n = (x-y)(x^{n-1} + x^{n-2}y +...+xy^{n-2} + y^{n-1})[/itex]. The first thing I noticed was that the term on the RHS side of the equation could possibly be expressed as an infinite series. And so, I undertook to find the general term:
I noticed that what was being summed was a product, the first term being [itex]x^{n-1}y^0[/itex], and the last [itex]x^0y^{n-1}[/itex], thus my general term needed to at least capture this.
Finally, [itex](x^{n-2} + x^{n-1}y +...+xy^{n-2} + y^{n-1}) = \sum_{i=0}^{n-1} x^{n-1-i}y^i[/itex]
[itex] \sum_{i=0}^{n-1} x^{n-1-i}y^i = x^{n-1} \sum_{i=0}^{n-1} x^{-i}y^i = x_{n-1} \sum_{i=0}^{n-1} (\frac{y}{x})^i[/itex]
This I knew to be the geometric series, the point at which I went wrong:
[itex]x_{n-1} \sum_{i=0}^{n-1} (\frac{y}{x})^i = x^{n-1} (\frac{1-(\frac{y}{x})^n}{1-\frac{y}{x}})[/itex]
I substituted in n rather than n-1. This, however, wasn't the most disconcerting thing, as I am sure you found it disconcerting: even with the incorrect substitution, I was able to properly solve the problem.
Why is that so? How can I fix this?
The problem to be solved is [itex]x^n - y^n = (x-y)(x^{n-1} + x^{n-2}y +...+xy^{n-2} + y^{n-1})[/itex]. The first thing I noticed was that the term on the RHS side of the equation could possibly be expressed as an infinite series. And so, I undertook to find the general term:
I noticed that what was being summed was a product, the first term being [itex]x^{n-1}y^0[/itex], and the last [itex]x^0y^{n-1}[/itex], thus my general term needed to at least capture this.
Finally, [itex](x^{n-2} + x^{n-1}y +...+xy^{n-2} + y^{n-1}) = \sum_{i=0}^{n-1} x^{n-1-i}y^i[/itex]
[itex] \sum_{i=0}^{n-1} x^{n-1-i}y^i = x^{n-1} \sum_{i=0}^{n-1} x^{-i}y^i = x_{n-1} \sum_{i=0}^{n-1} (\frac{y}{x})^i[/itex]
This I knew to be the geometric series, the point at which I went wrong:
[itex]x_{n-1} \sum_{i=0}^{n-1} (\frac{y}{x})^i = x^{n-1} (\frac{1-(\frac{y}{x})^n}{1-\frac{y}{x}})[/itex]
I substituted in n rather than n-1. This, however, wasn't the most disconcerting thing, as I am sure you found it disconcerting: even with the incorrect substitution, I was able to properly solve the problem.
Why is that so? How can I fix this?
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