# Higher Algebra Hall & Knight The General Theory of Continued Fractions (Chapter 31) Solutions

Hall and Knight Higher Algebra Solutions Chapter 31 ‘The General Theory of Continued Fractions’ has been created in order to guide you through the exercises based on the concepts given in this chapter of Higher Algebra by Hall and Knight. These solutions are a perfect guide for JEE and NEET competitive exams preparations.

Hall & Knight Algebra Mathematics book with solutions include a total of 33 questions divided into 2 exercises. The questions in this chapter are divided section-wise to help you learn more complex topics easily. There are word problems in the exercises besides the simple formula-based numerical problems. We recommend that you practice all the questions once for a strong base in the general theory of continued fractions.

The expert team of math teachers at Instasolv has formulated this set of Higher Algebra by Hall & Knight solutions fr Theory of Continued Fractions with great care. You will find the usage of simple and understandable language for explaining the general theory of continued fractions in a question and answer format. Step by step question-answer assistance is also provided with an elaborate description of each step. The solutions of this classic algebra book have been written in a format strictly compliant to the latest competitive exams guidelines for JEE and NEET.

## Important Topics Covered under Hall and Knight Higher Algebra Solutions Chapter 31: The General Theory of Continued Fractions

**Introduction**

The general form of Continued Fractions can be given assuch that a_{1}, a_{2}, a_{3},… , and b_{1}, b_{2}, b_{3 }can represent any values.

In this chapter, we will discuss the cases in which the sign before every component in the given continued fraction is positive, and the case in which the sign before every component is negative.

**Law of Formation of the Successive Convergents to the Continued Fraction, **

The first three convergents when the sign before every component is positive are

given as,

The (n+1)th convergent is given assuch thatand Similarly for

**Important Results**

- Assume the continued fractionThis fraction has a defined value if the limit of such that n is infinite is greater than zero.
- Assume the continued fractionif the denominator in each component is greater than the numerator by at least unity, then the convergents are positive fractions in ascending order of magnitude.
- Assume the continued fraction
**if each component in it is a proper fraction where numerator and denominator are integers, then the given continued fraction is incommensurable.** - A series can be converted into a continued fraction.

### Exercise Discussion of Hall and Knight Higher Algebra Solutions Chapter 31: The General Theory of Continued Fractions

- The exercise questions in this chapter comprehensively cover the general theory of continued fractions as well as the conversion of series into continued fractions.
- Exercise 31.a consists of 21 questions in which you will be required to find the convergents in the given continued fractions. You will also be required to find the product of two given continued fractions in certain questions using the general theory of continued fractions.
- The 12 questions in the exercise 31.b comprise 12 questions such that you will be required to convert given series into continued fractions or prove some important result which you may apply in the chapters ahead.

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