$$ a_{n} = 1 + \frac{1}{n^3} $$ Show that the sequence is converges $$ \lim_{n \rightarrow \infty} \left(1 + \frac{1}{1^3}\right)\left(1 + \frac{1}{2^3}\right)\left(1 + \frac{1}{3^3}\right) \ldots \left(1 + \frac{1}{n^3}\right) $$ I know that I should use natural logarithm but I have no clue how. Could you give me a hint?
2026-03-27 23:46:38.1774655198
Show that the sequence of products $\prod_{k=1}^n (1+1/k^3)$ converges
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Hint:
Note that $$\ln \left[ \left(1 + \frac{1}{1^3}\right)\left(1 + \frac{1}{2^3}\right)\left(1 + \frac{1}{3^3}\right) \ldots \left(1 + \frac{1}{n^3}\right)\right] = \ln \left(1 + \frac{1}{1^3}\right) + \cdots + \ln \left(1 + \frac{1}{n^3}\right) $$
Now use the inequality $\ln (1 + x) \leq x$ for all $x > 0$