Question 10

Let $$A=1^{-4}+2^{-4}+3^{-4}+\cdots$$, the sum of reciprocals of fourth powers of integers, and $$B=1^{-4}+3^{-4}+5^{-4}+\cdots$$, the sum of reciprocals of fourth powers of odd positive integers. The value of $$A/B$$ as a fraction is

Solution

The even terms in $$A$$ contribute $$\frac{1}{16}B$$ because $$\frac{1}{(2k)^4}=\frac{1}{16k^4}$$. Hence $$A=B+\frac{1}{16}B=\frac{17}{16}B$$ would incorrectly double-count the odd scaling, so instead write $$A=B+\frac{1}{16}A$$ since the even terms are exactly one sixteenth of all terms. Thus $$\frac{15}{16}A=B$$, giving $$A/B=16/15$$.

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Let $$A=1^{-4}+2^{-4}+3^{-4}+\cdots$$, the sum of reciprocals of fourth powers of integers, and $$B=1^{-4}+3^{-4}+5^{-4}+\cdots$$, the sum of reciprocals of fourth powers of odd positive integers. The value of $$A/B$$ as a fraction is

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