<s>[INST] Inequality $\frac{1}{1^3}+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{n^3}<\frac{3}{2}$ I'm trying to prove this by induction: $$\frac{1}{1^3}+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{n^3}<\frac{3}{2}, \quad n\geq1. $$
I got stuck in the inductive step:
$$\frac{3}{2} +\frac{1}{(n+1)^3}\,\ldots$$
Thank you for your help!
 [/INST] You can use for $n\geq 2$, $$\dfrac{1}{n^3} \leq \dfrac{1}{n(n+1)} = \dfrac{1}{n} - \dfrac{1}{n+1}$$
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