$\begin{aligned}11.\quad &\textrm{Diberikan}\\ &\quad A=1+\displaystyle \frac{1}{2^{\displaystyle 4}}+\displaystyle \frac{1}{3^{\displaystyle 4}}+\displaystyle \frac{1}{4^{\displaystyle 4}}+\displaystyle \frac{1}{5^{\displaystyle 4}}+\cdots\\ &\textrm{dan}\\ &\quad B=1+\displaystyle \frac{1}{3^{\displaystyle 4}}+\displaystyle \frac{1}{5^{\displaystyle 4}}+\displaystyle \frac{1}{7^{\displaystyle 4}}+\displaystyle \frac{1}{9^{\displaystyle 4}}+\cdots\\ &\textrm{Nyatakan}\quad \displaystyle \frac{A}{B}\quad \textrm{sebagai pecahan} \end{aligned}$
Jawaban: $\begin{aligned}&\textrm{Perhatikan bahwa}\\ &A=1+\displaystyle \frac{1}{2^{\displaystyle 4}}+\displaystyle \frac{1}{3^{\displaystyle 4}}+\displaystyle \frac{1}{4^{\displaystyle 4}}+\displaystyle \frac{1}{5^{\displaystyle 4}}+\displaystyle \frac{1}{6^{\displaystyle 4}}+\displaystyle \frac{1}{7^{\displaystyle 4}}+\cdots\\ &\textrm{Selanjutnya deret di ataskita bagi menjadi dua bagian, yaitu}:\\&A=\underset{\textrm{suku-suku genap}}{\underbrace{\left( \displaystyle \frac{1}{2^{\displaystyle 4}}+\displaystyle \frac{1}{4^{\displaystyle 4}}+\displaystyle \frac{1}{6^{\displaystyle 4}}+\displaystyle \frac{1}{8^{\displaystyle 4}}+\cdots \right)}}+\underset{\textrm{suku-suku ganjil}}{\underbrace{\left( 1+\displaystyle \frac{1}{3^{\displaystyle 4}}+\displaystyle \frac{1}{5^{\displaystyle 4}}+\displaystyle \frac{1}{7^{\displaystyle 4}}+\cdots \right)}}\\ &A=\displaystyle \frac{1}{2^{\displaystyle 4}}\left( 1+\displaystyle \frac{1}{2^{\displaystyle 4}}+\displaystyle \frac{1}{3^{\displaystyle 4}}+\displaystyle \frac{1}{4^{\displaystyle 4}}+\cdots \right)+\left( 1+\displaystyle \frac{1}{3^{\displaystyle 4}}+\displaystyle \frac{1}{5^{\displaystyle 4}}+\displaystyle \frac{1}{7^{\displaystyle 4}}+\cdots \right)\\ &A=\displaystyle \frac{1}{2^{\displaystyle 4}}A+B\Leftrightarrow A=\displaystyle \frac{1}{16}A+B\Leftrightarrow \displaystyle \frac{15}{16}A=B\Leftrightarrow \displaystyle \frac{A}{B}=\displaystyle \frac{16}{15}\\ &\textrm{Jadi, perbandingan}\quad \displaystyle \frac{A}{B}=\displaystyle \frac{16}{15} \end{aligned}$
Tidak ada komentar:
Posting Komentar
Informasi