Let \(S = \frac{1}{2^3} + \frac{1}{4^3} + \frac{1}{6^3} + \dotsb\) and \(T = \frac{1}{1^3} + \frac{1}{3^3} + \frac{1}{5^3} + \dotsb\). Find \(S/T\).
S=sumfor(n, 1, 1000, 1/(2*n)^3)=0.1502571129
T-sumfor(n, 1, 1000, 1/((2*n- 1 )^3))=1.05179979
S/T =0.1502571129 / 1.05179979
S/T = 1 / 7
+26367 help
Let
\(\large{S = \dfrac{1}{2^3} + \dfrac{1}{4^3} + \dfrac{1}{6^3} + \dotsb}\)
and
\(\large{T = \dfrac{1}{1^3} + \dfrac{1}{3^3} + \dfrac{1}{5^3} + \dotsb}\).
Find
\(\large{\dfrac{S}{T}}\)
1.
\(\mathbf{S+T =\ ?}\)
\(\begin{array}{|rcll|} \hline S+T &=& \left(\dfrac{1}{2^3} + \dfrac{1}{4^3} + \dfrac{1}{6^3} + \dotsb \right) +\left(\dfrac{1}{1^3} + \dfrac{1}{3^3} + \dfrac{1}{5^3} + \dotsb \right) \\ S+T &=& \color{red}{\dfrac{1}{1^3} + \dfrac{1}{2^3} + \dfrac{1}{3^3} + \dfrac{1}{4^3} + \dfrac{1}{5^3} + \dfrac{1}{6^3} + \dotsb} \\ \hline \end{array}\)
2.
\(\mathbf{\dfrac{S}{T} =\ ?} \)
\(\begin{array}{|rcll|} \hline S &=& \dfrac{1}{2^3} + \dfrac{1}{4^3} + \dfrac{1}{6^3}+ \dfrac{1}{8^3}+ \dfrac{1}{10^3} + \dotsb \\ &=& \dfrac{1}{(2\cdot 1)^3} + \dfrac{1}{(2\cdot 2)^3} + \dfrac{1}{(2\cdot 3)^3}+ \dfrac{1}{(2\cdot 4)^3}+ \dfrac{1}{(2\cdot 5)^3} + \dotsb \\ &=& \dfrac{1}{2^3 1^3} + \dfrac{1}{2^3 2^3} + \dfrac{1}{2^3 3^3}+ \dfrac{1}{2^3 4^3}+ \dfrac{1}{2^3 5^3} + \dotsb \\ &=& \dfrac{1}{2^3} \underbrace{ \left(\color{red}{\dfrac{1}{1^3} + \dfrac{1}{2^3} + \dfrac{1}{3^3}+ \dfrac{1}{4^3}+ \dfrac{1}{5^3} + \dotsb} \right) }_{=S+T} \\ &=& \dfrac{1}{2^3}(S+T) \\\\ S &=& \dfrac{1}{8}(S+T) \quad | \quad \cdot 8 \\ 8S &=& S+T \quad | \quad -S \\ 7S &=& T \quad | \quad :7 \\ S &=& \dfrac{T}{7} \quad | \quad:T \\ \mathbf{\dfrac{S}{T}} & \mathbf{=} & \mathbf{\dfrac{1}{7}} \\ \hline \end{array}\)
