求级数∑_n=1^∞(2n+1)/n²(n+1)²的和．

求级数∑_n=1^∞(2n+1)/n²(n+1)²的和．

因为(2n+1)/[n²(n+1)²]=(n²+2n+1-n²)/[n²(n+1)²] =1/n²-1/(n+1)²，所以s=lim_n→∞s_n =lim_n→∞[(1-1/2²)+(1/2²-1/3²)+…+(1/n²- 1/(n+1)²]=lim_n→∞(1-1/(n+1)²)=1．