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Infinite Sequences

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Define

$$A = \frac{1}{1^2} + \frac{1}{5^2} - \frac{1}{7^2} - \frac{1}{11^2} + \frac{1}{13^2} + \frac{1}{17^2} - \dotsb,$$

which omits all terms of the form $$1/{n^2}$$ where  is an odd multiple of 3, and

$$B = \frac{1}{3^2} - \frac{1}{9^2} + \frac{1}{15^2} - \frac{1}{21^2} + \frac{1}{27^2} - \frac{1}{33^2} + \dotsb,$$

which includes only terms of the form $$1/{n^2}$$ where $$n$$ is an odd multiple of 3.

Determine $$\frac{A}{B}.$$

If each sequence only had addition, this question would be much easier, but the subtraction makes it difficult for me.  Help is greatly appreciated!

Jun 4, 2021

#1
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Define

$$A=\dfrac{1}{1^2}+\dfrac{1}{5^2}-\dfrac{1}{7^2}-\dfrac{1}{11^2}+\dfrac{1}{13^2}+\dfrac{1}{17^2}-\dfrac{1}{19^2}-\dfrac{1}{23^2} + \dotsb,$$

which omits all terms of the form $$\dfrac{1}{n^2}$$where  is an odd multiple of 3, and

$$B=\dfrac{1}{3^2}-\dfrac{1}{9^2}+\dfrac{1}{15^2}-\dfrac{1}{21^2}+\dfrac{1}{27^2}-\dfrac{1}{33^2}+\dfrac{1}{39^2}-\dfrac{1}{45^2} + \dotsb,$$

which includes only terms of the form $$\dfrac{1}{n^2}$$ where  is an odd multiple of 3.

Determine $$\dfrac{A}{B}$$.

$$\begin{array}{|rcll|} \hline A&=&\dfrac{1}{1^2}+\dfrac{1}{5^2}-\dfrac{1}{7^2}-\dfrac{1}{11^2}+\dfrac{1}{13^2}+\dfrac{1}{17^2}-\dfrac{1}{19^2}-\dfrac{1}{23^2} + \dotsb \\ \hline B&=&\dfrac{1}{3^2}-\dfrac{1}{9^2}+\dfrac{1}{15^2}-\dfrac{1}{21^2}+\dfrac{1}{27^2}-\dfrac{1}{33^2}+\dfrac{1}{39^2}-\dfrac{1}{45^2} + \dotsb \\\\ 9B&=&\dfrac{1}{1^2}-\dfrac{1}{3^2}+\dfrac{1}{5^2}-\dfrac{1}{7^2} +\dfrac{1}{9^2}-\dfrac{1}{11^2} +\dfrac{1}{13^2}-\dfrac{1}{15^2} +\dfrac{1}{17^2}-\dfrac{1}{19^2}+\dfrac{1}{21^2} + \dotsb \\\\ 9B&=&A-\dfrac{1}{3^2}+\dfrac{1}{9^2}-\dfrac{1}{15^2}+\dfrac{1}{21^2}-\dotsb \\\\ 9B&=&A-\left( \underbrace{ \dfrac{1}{3^2}-\dfrac{1}{9^2}+\dfrac{1}{15^2}-\dfrac{1}{21^2}+\dotsb }_{=B}\right) \\\\ 9B&=&A-B \\ 10B &=& A \\ \mathbf{ \dfrac{A}{B} } &=& \mathbf{10} \\ \hline \end{array}$$ Jun 5, 2021