Express the following quantities as continued fractions and find the fourth convergent to each $.37$.

Hint: In this question we use the concept of convergent sequence that is one whose limit exists and is finite. So, first we write the successive quotient of the given fraction then write the series in continued fraction. Use this to find the fourth convergent.

Complete step-by-step answer:
According to the question we have, to Express the following quantities as continued fractions and to find the fourth convergent to each.
$.37$ $ = \dfrac{{37}}{{100}}$
$\therefore $The successive quotients are $2,1,2,2,1$
And this can be written as Continued fraction
$\dfrac{{37}}{{100}} = \dfrac{1}{{2 + }}\dfrac{1}{{1 + }}\dfrac{1}{{2 + }}\dfrac{1}{{2 + }}\dfrac{1}{{1 + }}\dfrac{1}{3};$
Here, the fourth convergent is $\dfrac{7}{{19}}$
Hence, fourth Convergent is $\dfrac {7} {{19}} $

Note: In mathematics, the limit of a sequence is the value that the terms of a sequence “tend to”. If such a limit exists, the sequence is convergence. It is always advisable to remember such concepts while involving convergent questions, as it saves a lot of time.