Find The First 5 Terms Of The Series Expansion Of 1 − ( X + 3 Y ) \frac{1}{-(x+3y)} − ( X + 3 Y ) 1 .

Mar 13, 2025 by ADMIN 104 views

**Series Expansion of a Rational Function**

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Introduction

In mathematics, a series expansion is a way to express a function as an infinite sum of terms. This is particularly useful for functions that cannot be expressed in a simple closed form. In this article, we will find the first 5 terms of the series expansion of the rational function $\frac{1}{-(x+3y)}$ .

Understanding the Function

The given function is $\frac{1}{-(x+3y)}$ . This is a rational function, meaning it is the ratio of two polynomials. The denominator is a linear polynomial, while the numerator is a constant polynomial.

Method of Series Expansion

To find the series expansion of the function, we will use the method of partial fractions. This involves expressing the function as a sum of simpler fractions, which can then be expanded using the binomial theorem.

Step 1: Express the Function as Partial Fractions

We can express the function as:

\frac{1}{-(x+3y)} = \frac{A}{x} + \frac{B}{3y}

where $A$ and $B$ are constants to be determined.

Step 2: Find the Constants A and B

To find the constants $A$ and $B$ , we can multiply both sides of the equation by the common denominator $-(x+3y)$ :

1 = -A(3y) - B(x)

Now, we can equate the coefficients of the terms on both sides of the equation:

-3Ay = 1

-Bx = 0

Solving for $A$ and $B$ , we get:

A = -\frac{1}{3y}

B = 0

Step 3: Express the Function as a Sum of Simpler Fractions

Now that we have found the constants $A$ and $B$ , we can express the function as a sum of simpler fractions:

\frac{1}{-(x+3y)} = -\frac{1}{3y}\left(\frac{1}{x}\right)

Step 4: Expand the Function using the Binomial Theorem

To expand the function using the binomial theorem, we can rewrite the term $\frac{1}{x}$ as:

\frac{1}{x} = x^{-1}

Now, we can apply the binomial theorem to expand the term:

x^{-1} = 1 - x + x^2 - x^3 + x^4 - \ldots

Step 5: Find the First 5 Terms of the Series Expansion

To find the first 5 terms of the series expansion, we can substitute the expanded term into the original function:

\frac{1}{-(x+3y)} = -\frac{1}{3y}\left(1 - x + x^2 - x^3 + x^4\right)

Now, we can simplify the expression to get the first 5 terms of the series expansion:

\frac{1}{-(x+3y)} = -\frac{1}{3y} + \frac{x}{3y} - \frac{x^2}{3y} + \frac{x^3}{3y} - \frac{x^4}{3y}

Conclusion

In this article, we have found the first 5 terms of the series expansion of the rational function $\frac{1}{-(x+3y)}$ . We used the method of partial fractions to express the function as a sum of simpler fractions, and then expanded the function using the binomial theorem. The first 5 terms of the series expansion are:

-\frac{1}{3y} + \frac{x}{3y} - \frac{x^2}{3y} + \frac{x^3}{3y} - \frac{x^4}{3y}

This result can be used to approximate the value of the function for small values of $x$ and $y$ .

Future Work

In future work, we can use the series expansion to approximate the value of the function for larger values of $x$ and $y$ . We can also use the series expansion to find the derivatives and integrals of the function.

References

[1] "Calculus" by Michael Spivak
[2] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Glossary

Series expansion: A way to express a function as an infinite sum of terms.
Partial fractions: A method of expressing a rational function as a sum of simpler fractions.
Binomial theorem: A formula for expanding a binomial expression.

Appendix

Proof of the Binomial Theorem: The binomial theorem can be proved using the method of mathematical induction.

Proof of the Binomial Theorem

The binomial theorem can be proved using the method of mathematical induction.

Let $n$ be a positive integer, and let $a$ and $b$ be real numbers. We want to prove that:

(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k

where $\binom{n}{k}$ is the binomial coefficient.

Base case: For $n=0$ , we have:

(a+b)^0 = 1 = \sum_{k=0}^0 \binom{0}{k} a^{0-k} b^k

Inductive step: Assume that the theorem is true for $n=k$ . We want to prove that it is true for $n=k+1$ .

We can write:

(a+b)^{k+1} = (a+b)^k (a+b)

Using the inductive hypothesis, we can expand the left-hand side:

(a+b)^{k+1} = \sum_{i=0}^k \binom{k}{i} a^{k-i} b^i (a+b)

Now, we can use the distributive property to expand the right-hand side:

(a+b)^{k+1} = \sum_{i=0}^k \binom{k}{i} a^{k-i} b^i a + \sum_{i=0}^k \binom{k}{i} a^{k-i} b^i b

Using the binomial theorem again, we can expand the first sum:

\sum_{i=0}^k \binom{k}{i} a^{k-i} b^i a = \sum_{i=0}^k \binom{k}{i} a^{k+1-i} b^i

Using the binomial theorem again, we can expand the second sum:

\sum_{i=0}^k \binom{k}{i} a^{k-i} b^i b = \sum_{i=0}^k \binom{k}{i} a^{k-i} b^{i+1}

Now, we can combine the two sums:

(a+b)^{k+1} = \sum_{i=0}^k \binom{k}{i} a^{k+1-i} b^i + \sum_{i=0}^k \binom{k}{i} a^{k-i} b^{i+1}

Using the definition of the binomial coefficient, we can rewrite the first sum:

\sum_{i=0}^k \binom{k}{i} a^{k+1-i} b^i = \sum_{i=0}^k \binom{k+1}{i} a^{k+1-i} b^i

Using the definition of the binomial coefficient, we can rewrite the second sum:

\sum_{i=0}^k \binom{k}{i} a^{k-i} b^{i+1} = \sum_{i=1}^{k+1} \binom{k}{i-1} a^{k-i+1} b^i

Now, we can combine the two sums:

(a+b)^{k+1} = \sum_{i=0}^k \binom{k+1}{i} a^{k+1-i} b^i + \sum_{i=1}^{k+1} \binom{k}{i-1} a^{k-i+1} b^i

Using the definition of the binomial coefficient, we can rewrite the first sum:

\sum_{i=0}^k \binom{k+1}{i} a^{k+1-i} b^i = \sum_{i=0}^{k+1} \binom{k+1}{i} a^{k+1-i} b^i

Using the definition of the binomial coefficient, we can rewrite the second sum:

\sum_{i=1}^{k+1} \binom{k}{i-1} a^{k-i+1} b^i = \sum_{i=0}^{k+1} \binom{k}{i} a^{k-i} b^i

Now, we can combine the two sums:

(a+b)^{k+1} = \sum_{i=0}^{k+<br/> # **Frequently Asked Questions (FAQs) about Series Expansion** =====================================================

Q: What is a series expansion?

A: A series expansion is a way to express a function as an infinite sum of terms. This is particularly useful for functions that cannot be expressed in a simple closed form.

Q: How do I find the series expansion of a function?

A: To find the series expansion of a function, you can use the method of partial fractions to express the function as a sum of simpler fractions, and then expand the function using the binomial theorem.

Q: What is the binomial theorem?

A: The binomial theorem is a formula for expanding a binomial expression. It states that: