# Show that the sequence of products $\prod_{k=1}^n (1+1/k^3)$ converges

$$a_{n} = 1 + \frac{1}{n^3}$$ 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?

• Is this the sequence {$(1+\frac{1}{n^3})^n$}? – Dunka Nov 21 '14 at 5:11
• Just for your curiosity, $a_{\infty}=\frac{\cosh \left(\frac{\sqrt{3} \pi }{2}\right)}{\pi }$. Nice, isn't it ? – Claude Leibovici Nov 21 '14 at 8:32
• – Lucian Nov 21 '14 at 9:09

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 $x > 0$
• Why prove that $$ln( \left(1 + \frac{1}{1^3}\right)\left(1 + \frac{1}{2^3}\right)\left(1 + \frac{1}{3^3}\right) \ldots \left(1 + \frac{1}{n^3}\right))$$ is converges implicate that the basic sequence is converges? – marrandior Nov 21 '14 at 5:21
• Because if we can show $\ln(a_n)$ converges, then $a_n$ converges. My hint can be used to show $\ln(a_n)$ is bounded above. It is also clearly increasing; hence it must have a limit. Therefore $a_n$ has a limit. – Simon S Nov 21 '14 at 5:23