Find the rational number, in lowest terms, given by each of the following continued fractions:
Find a simple continued fraction expansion for the following rational values:
Find the first several terms of the sequence associated with the simple continued fraction representation for $e$.
Find the first several terms of the sequence associated with the simple continued fraction representation for $\pi$.
The finite simple continued fractions $[1;2]$, $[1;2,2]$, $[1;2,2,2]$, $[1;2,2,2,2]$, $[1;2,2,2,2,2], \ldots $ are called the convergents of $[1;\overline{2}]$. More generally, the $n^{th}$ convergent for a given continued fraction is found by truncating the original continued fraction sequence to $n$ values.
First, find the exact value of $[1;\overline{2}]$. Then, to get a feel for how closely the convergents approximate their limiting value, find the difference between $[1;\overline{2}]$ and the value of each convergent listed.
Find the rational numbers, in lowest terms, given by each of the following continued fractions. Do you notice anything interesting? What value would have a simple continued fraction representation with an infinite string of $1$'s (i.e., $[1,\overline{1}]$)? Knowing that the continued fractions given below should better and better approximate $[1,\overline{1}]$, what might one conclude?
Find the continued fraction for $x=\displaystyle{\frac{22241739}{19848039}}$. Now compute the $\gcd(22241739,19848039)$ using the Euclidean Algorithm. Do you notice anything interesting? Make a conjecture, and test it against other examples.