Find out whether the series is convergent or divergent.

Question

∑^∞_n=0 1+n+n^(1/3) ÷ n+√(n⁴+n²+1)

Answer 1

i'm not entirely sure what expression you're trying to represent, but I'll guess you mean

∑ (1+n+n^1/3) / (n +√ [n⁴ + n² +1] )

by looking at it, i suspect that the series diverges. Unlucky for us, there are no singularities (aka infinities) in the expression (unless one considers complex numbers) so we will have to do some work.

Remember, if a smaller series diverges (as n goes to infinity), then a series larger than it will also diverge.

because n < (1+n+n^1/3) as n goes to ∞, if ∑ n/ (n +√ [n⁴ + n² +1] ) diverges so will the original expression

we also know that 1/ (n +√ [n⁴ + n² +1] ) > 1/( 4n²) as n goes to ∞, if ∑ n/ (4n² +1) diverges, so does the original expression.

(ignoring the n=0 term for simplicity, as this is just the test and its not needed when looking at infinity )

∑ n/ (4n² ) = ∑1/n -> diverges