Mathematics 2: Binomial Theorem, Power Series, and Linear Forms

Binomial Theorem

• Goal: To find an efficient method to calculate expressions of the form $(x + y)^n$, particularly for large values of $n$, without explicit multiplication.
• Degree of a Term: The sum of all occurring indices in a multiplicative term.
  • Examples: $x^2 y^2$ has degree $4$; $x^2 y$ has degree $3$.
  • In the expansion of $(x + y)^n$, every term has the degree $n$.
• Pascal’s Triangle: A geometric arrangement used to determine the coefficients of the binomial expansion.
  • Row 0: $1$
  • Row 1: $1, 1$
  • Row 2: $1, 2, 1$
  • Row 3: $1, 3, 3, 1$
  • Row 4: $1, 4, 6, 4, 1$
• General Formula: The binomial theorem is expressed as:   $(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k} = \binom{n}{0} x^0 y^{n-0} + \binom{n}{1} x^1 y^{n-1} + \dots + \binom{n}{k} x^k y^{n-k}$
• Binomial Coefficients: Defined as $\binom{n}{k} := \frac{n!}{(n - k)!k!}$, where $n! = n \times (n - 1) \times (n - 2) \times \dots \times 1$.

Sigma Notation

• Used to write sums in compact form: $\sum_{k=0}^{3} x_k = x_0 + x_1 + x_2 + x_3$.
• The index $k$ defines the number and form of the terms in the summation.

Power Series and Taylor Approximation

• Taylor Approximation: Approximating a function $f(x)$ using its derivatives at a specific point $x_0$:   $f(x) = f(x_0) + \frac{f'(x_0)}{1!} (x - x_0) + \frac{f''(x_0)}{2!} (x - x_0)^2 + \dots = \sum_{k=0}^{\infty} \frac{f^{(k)}(x_0)}{k!} (x - x_0)^k$
• Maclaurin Series: A Taylor expansion centered at $x_0 = 0$.
  • Example: $\cos(x) = 1 - \frac{x^2}{2!} + \dots$
  • Example: $e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}$

Convergence and Ratio Test

• Convergence: A Taylor series converges if its terms decline to zero at a specific rate.
• Ratio Test: A series $\sum_{k=0}^{\infty} a_k$ converges absolutely if:   $\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = L < 1$
  • If $L = 0$, the series converges for all $x$.
  • If $L$ is infinite, it only converges at $x = a$.
  • If $L$ is finite and non-zero, it converges for $|x - a| < \frac{1}{L}$.

Solving Simultaneous Equations

• Methods:
  • Substitution: Express one variable in terms of another and plug it into the second equation.
  • Elimination: Linearly combine equations (add, subtract, or use multiples) to reduce the system to a single unknown.
• Possible Outcomes:
  • Unique Solution: The lines intersect at exactly one point.
  • Infinitely Many Solutions: The equations are multiples of each other (the same line).
  • No Solution: The graphs are parallel and do not intersect.

Determination of Law (Linear Forms)

• Concept: Modifying non-linear equations into the linear form $Y = mX + c$ to determine constants from experimental data.
• Transformation Examples:
  • $y = ax^2 + b \rightarrow$ Plot $Y = y$ against $X = x^2$.
  • $y = a \left(\frac{1}{x}\right) + b \rightarrow$ Plot $Y = y$ against $X = \frac{1}{x}$.
  • $y = ax^2 + bx \rightarrow$ Plot $Y = \frac{y}{x}$ against $X = x$, where $\frac{y}{x} = ax + b$.
  • $y = ax^2 + bx + k \rightarrow$ Plot $Y = \frac{y - k}{x}$ against $X = x$.

Questions & Discussion

• Question: How would we calculate expressions where $n$ is a rational number (not an integer), e.g. $(x + y)^{\frac{1}{2}}$?
  • Response: This is addressed through Power Series expansions.
• Question: Is it always possible to determine two unknowns with two equations?
• Question: Does this generalize to more unknowns, e.g. solve three unknowns with three equations?