Consider the series $$\sum_{n=3}^\infty \frac{a^n}{n^b (\log_e n)^c}.$$​​

For which values of $$a, b, c \in \mathbb{R}$$​, does the series $$\textbf{NOT converge}?$$​​

Correct option is C

Consider,

$$U_n = \sum_{n=3}^\infty \frac{a^n}{n^b (\log_e n)^c}.$$

​

If  |a| > 1 , the exponential term  $$a^n$$​  grows faster than any polynomial or logarithmic decay, so  $$U_n$$​ will diverge to  $$\infty$$​.

If  |a| = 1, then convergence of  $$U_n$$​ depends upon  $$n^b$$​  and  $$(log_e n)^c.$$​

For  |a| < 1, exponential decay ensures convergence.

​$$\textbf{NOTE:}$$​ From structure of series, we can conclude that  b > 1  and  c > 1  can make the series convergent due to p - test and comparison test.
So let's test $$\textbf{Option C:}$$​​

​$$a = +1 , 0 \leq b \leq 1 , 0 < c < 1$$

Let  b = c =$$\frac{1}{2}$$​​:

​$$U_n = \sum \frac{a^n}{n^{1/2} \sqrt{\log n}} \quad \text{which is divergent.}\\[10pt]\implies \text{Option C is correct.}$$​​