How do I evaluate Lim n→∞ (1+2+3+…+n^3) /n^4?

Answer 1

1 +2+3+...+(n^3-1)+n^3=(1+n^3)(n^3)/2 by using the Gauss formula for the sum of consecutive integers, note that this is not a polynomial, just sum of consecutive integers.

Then it is equal to lim (1+n^3)(n^3)/2/n^4=lim(n^3/(2n^4))+(n^6/(2n^4))=lim1/(2n)+lim(n^2/2)-->0+∞=∞

Btw, Gauss formula 1+2+3+...+(m-1)+m=m(m+1)/2

Answer 2

Notice 1 + 2 + ... + n³ is a polynomial and n⁴ is a polynomial. Then to find lim_n→∞^{(1 + 2 + ... + n^3)}/_n^4, we only need to look at the leading terms n³ and n⁴:

lim_n→∞^{n^3}/_n^4 = lim_n→∞¹/_n = 0

Therefore, lim_n→∞^{(1 + 2 + ... + n^3)}/_n^4 = 0.