5i^3/n^8 if i = 1. Rewrite rational function

Question

Rewrite as a rational function and find infinity limit S(n)

Answer 1

It's not clear what 5i³/n⁸=(5/n⁸)(i³) means, so I'll assume that you're referring to the sum of a series:

(5/n⁸)∑i³ where integer i: 1≤i≤n.

That is S_n=(5/n⁸)(1+8+27+...+(n-1)³+n³)=(5/n⁸)(n(n+1)/2)².

S_n=5(n+1)²/(4n⁶).

INDUCTIVE PROOF

If sum of cubes of the natural numbers is S_n=(n(n+1)/2)², then:

S_n+1=S_n+(n+1)³. S₁=1, and S₂=9=(2×3/2)², so base case is true.

S_n+1 should be ((n+1)(n+2)/2)²=(n²+3n+2)/4=(n⁴+6n³+13n²+12n+4)/4.

S_n+(n+1)³=(n⁴+2n³+n²)/4+n³+3n²+3n+1=

(n⁴+2n³+n²+4n³+12n²+12n+4)/4=(n⁴+6n³+13n²+12n+4)/4=S_n+1. This proves by induction that the formula used is correct.