Vtyjkyuk

this link contain a limit related math of calculus.generally when x tends to infinity we take the highest power as common but in this math the highest power is not same in dividend and the divisor.plz help me out to sove it.i am just a beginner in calculus.so plz accept my apology if the question is too silly and my idea to solve the question is wrong

$$lim_xtoinftyfrac1^2+2^2+3^2+cdots+x^2x^3+x^2+x+1$$

HINT

Recall that

$$sum_j=1^x j^2=fracx(x+1)(2x+1)6$$

and more in general by Faulhaber's formula

$$sum_j=1^x j^k sim fracx^k+1k+1$$

Assume that the numerator can be put in the form of a polynomial, let $P(x)$. From its definition we have

$$P(x)-P(x-1)=x^2.$$

As the first order difference of a polynomial is a polynomial one degree less, $P$ must be cubic, let

$$P(x)=ax^3+bx^2+cx+d.$$

Now from

$$P(x)-P(x-1)=a(x^3-x^3+3x^2-3x+1)+b(x^2-x^2+2x-1)+c(x-x+1)=x,$$

we draw $3a=1$. The other coefficients could be computed but we don't need them. It suffices to notice that the system is triangular so it does have a solution and the representation as a polynomial holds.

The limit is $$frac13.$$