Question

what is the sum of the series 1-x+x²-x³...-x^(2n-1)​

Answer 1

Answer:

hope my answer helps you......

Answer 2

$S_N=1-x+x^2-x^3+\ldots-x^{2n-1}\\S_N=(1+x^2+x^4+\ldots+x^{2n-2})-(x+x^3+x^5+\ldots+x^{2n-1})\\\\{S_N}_1=1+x^2+x^4+\ldots+x^{2n-2}\\{S_N}_2=x+x^3+x^5+\ldots+x^{2n-1}\\\\\underline{{S_N}_1}\\a_1=1\\q=x^2\\{S_N}_1=1\cdot\dfrac{1-(x^2)^n}{1-x^2}=\dfrac{1-x^{2n}}{1-x^2}=\dfrac{x^{2n}-1}{x^2-1}\\\\\underline{{S_N}_2}\\a_1=x\\q=x^2\\{S_N}_2=x\cdot\dfrac{1-(x^2)^n}{1-x^2}=\dfrac{x(1-x^{2n})}{1-x^2}=\dfrac{x(x^{2n}-1)}{x^2-1}$

$S_N=\dfrac{x^{2n}-1}{x^2-1}-\dfrac{x(x^{2n}-1)}{x^2-1}\\\\S_N=\dfrac{x^{2n}-1-x(x^{2n}-1)}{x^2-1}\\\\S_N=\dfrac{(x^{2n}-1)(1-x)}{x^2-1}\\\\S_N=-\dfrac{(x^{2n}-1)(x-1)}{x^2-1}\\\\S_N=-\dfrac{x^{2n}-1}{x+1}$