ow that (a bn > In (1+ ) (1 + ±) P2 c) Show that: limn->00 (1+ (1+±)· (1 + ) P2
For every natural nonzero number, $n \in\mathbb{N}^{*}$, we note $P_{n}$ the nth prime number. Also we Consider the sequences
$$
\begin{array}{l}
a_{n}=\frac{p_{1}}{p_{1}-1} \cdot \frac{p_{2}}{p_{2}-1} \cdots \cdot \frac{p_{n}}{p_{n}-1} \\
b_{n}=\frac{1}{p_{1}}+\frac{1}{p_{2}}+\cdots+\frac{1}{p_{n}}
\end{array}
$$
a) Show that $\lim _{n\rightarrow \infty} a_{n}=\infty$
b) SHow that $b_{n}>\ln \left(1+\frac{1}{p_{1}}\right) \cdot\left(1+\frac{1}{p_{2}}\right) \cdots \cdot\left(1+\frac{1}{p_{n}}\right)$
c) Show that $\lim _{n \rightarrow \infty}\left(1+\frac{1}{p_{1}}\right) \cdot\left(1+\frac{1}{p_{2}}\right) \cdots \cdot\left(1+\frac{1}{p^{n}}\right)=\infty$ and calculate the $lim_{n\rightarrow \infty} b_{n}.$
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