How to prove this series inequality by mathematical induction It is known that an + 1 (refers to the N + 1st term) = an + (an ^ 2) / (n ^ 2), A1 = 1 / 3. To prove an > 1 / 2 - 1 / 4N. In addition, I will reduce the inequality to a more strict one, and it is better to prove an > 1 / 2 - 1 / 5N first. What is the reason?

How to prove this series inequality by mathematical induction It is known that an + 1 (refers to the N + 1st term) = an + (an ^ 2) / (n ^ 2), A1 = 1 / 3. To prove an > 1 / 2 - 1 / 4N. In addition, I will reduce the inequality to a more strict one, and it is better to prove an > 1 / 2 - 1 / 5N first. What is the reason?

prove:
When n = 1, A2 = a1 + (A1 ^ 2) / 1 ^ 2 = 1 / 3 + 1 / 18 = 7 / 18 > 1 / 2-1 / 4 = 7 / 28 holds
Let n = k, AK + 1 = AK + (AK ^ 2) / (k ^ 2) > 1 / 2-1 / 4K
When n = K + 1, AK + 2 = AK + 1 + (AK + 1 ^ 2) / (K + 1) ^ 2
>1/2-1/4k+(1/2-1/4k)^2/(k+1)^2
=1/2-1/4[k-(1/2-1/4k)^2/(k+1)^2]
=1/2-1/4[(k^3+2k^2+k-1+k-1/4k^2)/(k+1)^2]
>1/2-1/4[(k^3+3k^2+3k+1)/(k+1)^2]
=1/2-1/4[(k+1)^3/(k+1)^2]
=1 / 2-1 / 4 (K + 1) set up
So an > 1 / 2 - 1 / 4N