Convergence or divergence

real-analysis convergence-divergence limits

If you don't want to resort to the "big mallets" of the Ratio Test or the Limit Comparison Test, you could also note that

$$ \frac{n + 2^n}{n + 3^n} \ < \ \frac{2 \ \cdot \ 2^n}{n + 3^n} \ < \ \frac{2 \ \cdot \ 2^n}{ 3^n} \ , \ \text{for} \ n \ \ge \ 1 \ , $$

the last term in this inequality being the general term of a convergent geometric series.

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