Question

求和:3/1×2×4+5/2×3×5+7/3×4×6+....+(2n+1)/n(n+1)(n+3)

Answer 1

因为2N+1=N+(N+1)，
所以(2N+1)/N(N+1)(N+3)=[N+(N+1)]/N(N+1)(N+3)=1/(N+1)(N+3) + 1/N(N+3)
=1/2 *[1/(N+1) - 1/(N+3) ] + 1/3 *[1/N - 1/(N+3)]
故原式可化为：
1/2 (1/2 -1/4)+1/3(1-1/4) +1/2(1/3-1/5)+1/3(1/2-1/5)+1/2(1/4-1/6)+1/3(1/3-1/6)+…………
=1/2(1/2-1/4+1/3-1/5+1/4-1/6+1/5-1/7+……)+1/3(1-1/4+1/2-1/5+1/3-1/6+1/4-1/7+1/5-1/8+……)
=1/2[1/2+1/3-1/(N+2)-1/(N+3)]+1/3[1+1/2+1/3-1/(N+1)-1/(N+2)-1/(N+3)]
=37/36-1/(3N+3)-5/(6N+12)-5/(6N+18)