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I need to use the comparison test to determine whether the series:

$\sum\frac{3^n+7^n}{3^n+8^n}$

converges or diverges.

So far, I've tried splitting it into $\frac{3^n}{3^n + 8^n} + \frac{7^n}{3^n + 8^n} $ but have gotten nowhere

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    $\begingroup$ The first one is dominated by $\left(\frac 38\right)^n$. The second is dominated by $\left(\frac 78\right)^n$. $\endgroup$
    – lulu
    Nov 20, 2020 at 22:20
  • $\begingroup$ I see now. Thank you $\endgroup$
    – user842286
    Nov 20, 2020 at 22:24

2 Answers 2

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hint

For the numerator, $$3^n+7^n\le 2.7^n$$

For the denominator, $$3^n+8^n\ge 8^n$$

thus

$$0<\frac{3^n+7^n}{3^n+8^n}\le 2\frac{7^n}{8^n}$$

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It will be simpler to use the limit comparison test via *asymptotice equivalence of functions: $$3^n+7^n\sim_\infty 7^n,\quad3^n+8^n\sim_\infty 8^n,\quad\text{ hence }\quad \frac{3^n+7^n}{3^n+8^n}\sim_\infty \frac{7^n}{8^n}=\Bigl(\frac78\Bigr)^n,$$ which converges.

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