Show that this sum is divergent

Question

How can I argue that this sum i divergent?

$$\sum_{j=1}^\infty \left( \frac{1}{j\beta}+\frac{\delta}{(j-1)\beta^2}+ \frac{\delta^2}{(j-2)\beta^3}+ \cdots + \frac{\delta^{j-1} }{\beta^j} \right)$$

Here $\beta,\delta>0$ are constants.

I hope someone can help me or give me a hint.

Edit: I hope the expression of the sum is correct. I am trying to write up the sum $$\sum_{j=1}^\infty \left( \frac{1}{\beta_j}+\frac{\delta_j}{\beta_j\beta_{j-1}}+ \cdots + \frac{\delta_j \cdots \delta_1 }{\beta_j\cdots \beta_0} \right)<\infty.$$

where $\beta_j=j\beta$ and $\delta_j=j\delta$.

Answer 1

$$\sum_{j=1}^{+\infty}\frac{1}{j\beta}$$ is not convergent, hence the whole series cannot be convergent.