I am trying to study the convergence of the piecewise sum $sum_{n=1}^infty a_n$ where

$$a_n =

begin{cases}

(1+frac{1}{43n})^{n^{43}}, & n=3p, p in mathbb{N}^{*} \

7*5^n & n=3p+1, p in mathbb{N} \

33 n, & n=3p+2, p in mathbb{N} \

(frac18)^n & n = 3p+3, p in mathbb{N}

end{cases}$$

It is obvious by the divergence test that all cases diverge, except $(frac18)^n $ which converges as a geometric sum.

But to me, it is not obvious, what $sum_{n=1}^infty a_n$ does.

**Note:** My intuition tells me that since there is **at least one case** where the sum goes to infinity (diverges) then no matter what the other cases do, the sum is going to diverge ($sum_{n=1}^infty a_n= +infty$). But I cannot prove it.

So, does the sum converge or diverge, and why?