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

$\displaystyle\sum_{n=1}^\infty\frac{n+1}{n^9+n}$

B: Since

$\dfrac{n+1}{n^9+n}=\dfrac{n+1}{n(n^8+1)}\approx\dfrac1{n^8}$

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)

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

$\implies\dfrac{n+1}{n^9+n}<\dfrac1{n^7}$