[tex] \frac{y}{x} [/tex]
adalah....
Nilai dari y/x adalah 31/32.
Pembahasan
Deret Harmonik
Diketahui
[tex]\begin{aligned}&\bullet\ x=\frac{1}{1^5}+\frac{1}{2^5}+\frac{1}{3^5}+\frac{1}{4^5}+\frac{1}{5^5}+{\dots}\\&\bullet\ y=\frac{1}{1^5}+\frac{1}{3^5}+\frac{1}{5^5}+\frac{1}{7^5}+\frac{1}{9^5}+{\dots}\\\end{aligned}[/tex]
Ditanyakan
[tex]\dfrac{y}{x}=\dots[/tex]
PENYELESAIAN
[tex]\begin{aligned}x&=\frac{1}{1^5}+\frac{1}{2^5}+\frac{1}{3^5}+\frac{1}{4^5}+\frac{1}{5^5}+{\dots}\end{aligned}[/tex]
[tex]\begin{aligned}x&=\overbrace{\frac{1}{1^5}+\frac{1}{3^5}+\frac{1}{5^5}+\frac{1}{7^5}+{\dots}}^{\begin{array}{c}\textsf{suku-suku ganjil}\end{array}}\\&\quad+\underbrace{\frac{1}{2^5}+\frac{1}{4^5}+\frac{1}{6^5}+\frac{1}{8^5}+{\dots}}_{\begin{array}{c}\textsf{suku-suku genap}\end{array}}\end{aligned}[/tex]
[tex]\begin{aligned}x&=\overbrace{\frac{1}{1^5}+\frac{1}{3^5}+\frac{1}{5^5}+\frac{1}{7^5}+{\dots}}^{\begin{array}{c}y\end{array}}\\&\quad+\frac{1}{2^5}\underbrace{\left(\frac{1}{1^5}+\frac{1}{2^5}+\frac{1}{3^5}+\frac{1}{4^5}+{\dots}\right)}_{\begin{array}{c}x\end{array}}\end{aligned}[/tex]
[tex]\begin{aligned}x&=y+\frac{x}{2^5}=\frac{2^5y+x}{2^5}\\x&=\frac{32y+x}{32}\\32x&=32y+x\iff31x=32y\\\therefore\ \frac{y}{x}&=\boxed{\:\bf\frac{31}{32}\:}\end{aligned}[/tex]
[tex]\blacksquare[/tex]
