Simplifying $1 - x + x^2 - x^3 + ... + x^{98} - x^{99}$ to an equivalent expression.
I am doing an exercise to see the error when solving this polynomial for $x = 1.00001$ using nested multiplication.
I believe the correct way to achieve this simplification (based on a lecture) is to multiply the polynomial by $\frac{1+x}{1+x}$; however, my algebra skills are not really up to par so I am failing to see the purpose of this. I assume it is to cancel out most of the terms in the polynomial - but which ones are being canceled?
$\endgroup$3 Answers
$\begingroup$Here's the beginning:
$$\begin{align} &\ \ \ 1-x+x^2-x^3+\cdots+ x^{98}-x^{99}\\ &= \left(1-x+x^2-x^3+\cdots+ x^{98}-x^{99}\right)\frac{1+x}{1+x}\\ &= \frac{1(1+x)-x(1+x)+x^2(1+x)+\cdots +x^{98}(1+x)-x^{99}(1+x)}{1+x}\\ &=\frac{1+x-x-x^2+x^2+x^3-x^3\cdots +x^{98}+ x^{99}-x^{99} -x^{100}}{1+x}\\ \end{align}$$
Can you see where this goes?
$\endgroup$ 1 $\begingroup$this is a geometric sequence i.e.$(-x)^0+...+(-x)^{99}$,using the sum or geometric seuqence,i.e.$1+r+...+r^n=\frac{r^{n+1}-1}{r-1}$,so in this case put r=-x n=99 and you get the expression
$\endgroup$ 0 $\begingroup$You could try to substitute x by another variable, for instance let x = -y in the formula, and see what happens...
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