Question

当x的绝对值小于1,求(1+x)(1+x^2)(1+x^4)…(1+x^2^n)n趋近于无穷的极限

Answer 1

(1+x)(1+x^2)(1+x^4)…(1+x^2^n)n

=(1-x)(1+x)(1+x^2)(1+x^4)...(1+x^2^n)/(1-x)

=(1-x^2)(1+x^2)(1+x^4)...(1+x^2^n)/(1-x)

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

lim(n->∞)[(1+x)(1+x^2)(1+x^4)…(1+x^2^n)n]

=lim(n->∞)[(1-x^2(n+1))/(1-x)]

=1/(1-x)