`(1-x)(1+x+x^(2))`

For x>0 the sum of the series (1)/(1+x)-(1-x)/((1+x)^(2))+((1-x)^(2))/((1+x)^(3))-...oo is equal to

lim_(x rarr1)((1-x)(1-x^(2))............(1-x^(2n)))/({(1-x)(1-x^(2)).........(1-x^(n))}^(2))

lim_(xto1) ((1-x)(1-x^(2))...(1-x^(2n)))/({(1-x)(1-x^(2))...(1-x^(n))}^(2)), n in N , equals

lim_(xto1) ((1-x)(1-x^(2))...(1-x^(2n)))/({(1-x)(1-x^(2))...(1-x^(n))}^(2)), ninN," equals"

lim_(x rarr0)((1)/(x sin^(-1)x)-(1-x^(2))/(x^(2)))

For x gt 0 the sum of the series (1)/(1+x) - ((1-x))/((1+x)^(2))+ ((1-x)^(2))/((1+x)^(3))- cdots oo is equal to

int(ln(In((1+x)/(1-x))))/(1-x^(2))dx

((x+1)/(x^(2/3)-x^(1/3)+1)-(x-1)/(x-x^(1/2)))=

int(e^(tan^-1(x)))/(1+x^(2))(1+x+x^(2))dx=...+c