First slide

Special Series in Sequences and Series

Question

$Let p = 1^{4} + 2^{4} + 3^{4} + 4^{4} + \dots \infty and q = 1^{4} + 3^{4} + 5^{4} + 7^{4} + \dots \infty then \frac{p}{q} is$

Moderate

A

$\frac{16}{15}$
B

$\frac{15}{16}$
C

$\frac{5}{6}$
D

$\frac{6}{7}$

Solution

$\begin{array}{l} p = \frac{1}{1^{4}} + \frac{1}{2^{4}} + \frac{1}{3^{4}} + \dots \infty \\ = (\frac{1}{1^{4}} + \frac{1}{3^{4}} + \frac{1}{5^{4}} + . .) + \frac{1}{2^{4}} (\frac{1}{1^{4}} + \frac{1}{2^{4}} + \frac{1}{3^{4}} + . .) \\ p = q + \frac{1}{16} p \end{array}$
$\frac{15}{16} p=q \to \frac{p}{q} = \frac{16}{15}$

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