I have to find the radius of convergence of the series $$\sum_{n=0}^{\infty} (3^n + (-5)^n) x^{7n}$$ I know that I will have something like $|x^7|<\frac{1}{L}$. I tried finding $R$ with $$\lim_{n\rightarrow\infty} \lvert \frac{a_{n+1}}{a_{n}}\rvert$$ but I have trouble finding that limit.I just can't figure it out.So any ideas and solutions on finding the limit are welcomed $\ddot\smile $
2026-04-05 18:40:38.1775414438
Find the radius of convergence of $\sum_{n=0}^{\infty} (3^n + (-5)^n) x^{7n}$
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It's $|\frac{a_{n+1}}{a_n}|$. Taking the ratio $|\frac{3^{n+1} +(-5)^{n+1}}{3^n + 5^n}|$ and dividing both denominator and numerator by $5^{n+1}$ you see it converges to $5$, so you have $ |x^7| <\frac{1}{5}$.