Question

find the fractional equivalent of 0.41717....

Answer 1

Try to write it as
\[
\frac{4}{10}+\frac{17 \sum _{n=0}^{\infty } \frac{1}{10^{2 n}}}{10^3}
\]

Answer 2

\[0.41717=\frac{4}{10}+\frac{17}{10^3}+\frac{17}{10^5}+\frac{17}{10^7}+\cdots

\]

Answer 3

\[
0.41717=\frac{4}{10}+\frac{17}{10^3}+\frac{17}{10^5}+\frac{17}{10^7}+\cdots=\\
\frac{4}{10}+\frac{17}{10^3}\left(1+\frac{1}{10^2}+\frac{1 }{10^4}+\cdots\right)=
\]

Answer 4

Use geometric series methods to show that
\[
\sum _{n=0}^{\infty } \frac{1}{10^{2 n}}=\frac{100}{99}
\]

Answer 5

\[
0.41717\cdots=\frac{4}{10}+\frac{17}{10^3}+\frac{17}{10^5}+\frac{17}{10^7}+\cdots=\\
\frac{4}{10}+\frac{17}{10^3}\left(1+\frac{1}{10^2}+\frac{1 }{10^4}+\cdots\right)=
\frac{4}{10}+\frac{17}{10^3}\sum _{n=0}^{\infty } \frac{1}{10^{2 n}}=\\

\frac{4}{10}+\frac{17}{10^3}\frac{100}{99}=\frac {413}{990}
\]

Answer 6

maybe easier like this :
let x=0.41717... (1)
10x=4.1717... (2)
1000x=417.17... (3)
(3)-(2) :
990x=413
x=413/990