Does $$\sum_{n=1} ^{\infty} \frac{5^{n}-2^{n}}{7^{n}-6^{n}}$$
converge? I tried the ratio test but I failed.
2026-03-30 07:00:22.1774854022
Determine the convergence of the series $\sum_{n=1} ^{\infty} \frac{5^{n}-2^{n}}{7^{n}-6^{n}}$
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Ratio test: $$ \frac{\displaystyle\frac{5^{n+1}-2^{n+1}}{7^{n+1}-6^{n+1}}}{\displaystyle\frac{5^{n}-2^{n}}{7^{n}-6^{n}}} =\frac{7^{n}-6^{n}}{{7^{n+1}-6^{n+1}}}\,\frac{5^{n+1}-2^{n+1}}{5^{n}-2^{n}} =\frac{1-(6/7)^n}{7-(6/n)^{n+1}}\,\frac{5-(2/5)^{n+1}}{1-(2/5)^n}\to\frac57<1, $$ so the series converges.