<s>[INST] Convergence of a sequence $\frac{1}{1+n^3}$ How can I prove by integral test that the sequence $1+ \dfrac{1}{1+2^3} + \dfrac{1}{1+3^3} + \dots + \dfrac{1}{1+n^3}$ is convergent?
Thank you. Is there a way that I can integrate $\dfrac{1}{1+n^3}$ ?
 [/INST] Just in case somebody doesn't feel like doing calculus today:
$$\sum_{n=2}^\infty \frac{1}{1+n^3} < \frac{1}{2} \sum_{n=2}^\infty \frac{2}{n^3-n} = \frac{1}{2} \sum_{n=2}^\infty \left(\frac{1}{(n)(n-1)} - \frac{1}{(n+1)(n)}\right) = \frac{1}{2} \cdot \frac{1}{2(2-1)} = \frac{1}{4}$$
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