I was having some error calculating this question. I was wondering if anyone could help me out!
Solve for x:
$\sum_{n=1}^\infty 3x^{5n} = 27$
2026-05-05 10:13:25.1777976005
Solving Summation Expression
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$$\sum_{n=1}^{+\infty}3x^{5n}=3\sum_{n=1}^{+\infty}(x^5)^n=\frac{3x^5}{1-x^5}$$ for all $|x|<1$ (otherwise the series diverges). Thus $$ \sum_{n=1}^{+\infty}3x^{5n}=27\iff x^5=9(1-x^5)\iff x=\left(\frac{9}{10}\right)^{1/5} $$