Why does this product diverge?

Question

Consider the partial product $$ p_n=\prod_{k=1}^{2n}a_k,$$ where $$a_k=\left\{\begin{array}{ll} k & \text{for }k\text{ odd} \\ \frac{1}{k-1} & \text{for }k \text{ even} \end{array}\right..$$ Clearly $p_n=1$ for all natural numbers $n$. But, according to comments left in an earlier post the limit of partial sums as $n\to\infty$ diverges. Why is this so?

Answer 1

As you noted correctly,

$$\prod_{k=1}^{2n} a_k = 1$$

but

$$\prod_{k=1}^{2n+1} a_k = 2n+1$$

which converges to $\infty$ as $n\rightarrow\infty$.

Therefore the sequence of partial products does not converge, i.e. the infinite product

$$\prod_{k=1}^\infty a_k$$

does not exist, that is, it diverges.