幂级数∑_n=1^∞[1/(n+n²)](x-1)ⁿ的收敛区间为____.

幂级数∑_n=1^∞[1/(n+n²)](x-1)ⁿ的收敛区间为____.

(0，2)。解析：由于lim_n→∞=1/[n+1+(n+1)²]/[1/(n+n²)]=lim_{n→∞2)/[n+1+(n+1)²]limn→∞=(1+1/n)/[(1/n+1/n²)+(1+1/n)²]所以R=1，收敛区间为-1﹤x-1﹤1，即(0，2)．}