<s>[INST] Induction proof done, need to simplify $\sqrt[4]{(n+1)^5}\ln(1 + \frac{1}{(n+1)^2}) \leq 4 \sqrt[4]{\frac{n+1}{n}}$ Prove that:
$$\sum_{j=1}^{n} \sqrt[4]{j^5}\ln(1+\frac{1}{j^2}) \leq 4 \sqrt[4]{n}, n \in \mathbb{N} $$
I got everything right and came up with an inequality:
$$\sqrt[4]{(n+1)^5} \ln\left(1 + \frac{1}{(n+1)^2}\right) \leq 4  \sqrt[4]{\frac{n+1}{n}}$$
The problem is I cannot simplify that to get my solution all done. Anybody got an idea?
 [/INST] As regards your final step, just notice that $\ln(1+x)\leq x$.Therefore it suffices to show that
$$\sqrt[4]{(n+1)^5}\cdot \frac{1}{(n+1)^2} \leq 4  \sqrt[4]{\frac{n+1}{n}}$$
that is
$$\sqrt[4]{n}\leq 4 (n+1)^{\frac{1}{4}+2-\frac{5}{4}}=4(n+1)$$
which trivially holds.
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