Question

Determine whether the series converges or diverges.

(n=1→[infinity]) Σ (n + 1)/ (n⁹ + n)

a) The series converges by the Comparison Test. Each term is less than that of a convergent geometric series.
b) The series converges by the Comparison Test. Each term is less than that of a convergent p-series.
c) The series diverges by the Comparison Test. Each term is greater than that of a divergent p-series.
d) The series diverges by the Comparison Test. Each term is greater than that of a divergent harmonic series.

Answer 1

`\displaystyle\sum_(n=1)^\infty(n+1)/(n^9+n)`

B: Since

`(n+1)/(n^9+n)=(n+1)/(n(n^8+1))\approx\frac1{n^8}`

we can compare the given series to the convergent p-series,

`\displaystyle\sum_(n=1)^\infty\frac1{n^7}`

We have

`n^8<n^9`

`\implies n^8+n^7<n^9+n`

(for large enough
`n`, the degree-9 polynomial dominates the degree-8 polynomial)

`\implies n^7(n+1)<n^9+n`

`\implies(n+1)/(n^9+n)<\frac1{n^7}`