Binomial Theorem

Exercise 1 : Expansion

Expand using the binomial formula:

1. $ (x + y)^4 $
2. $ (2a - b)^5 $
3. $ \left( \frac{1}{x} + x \right)^3 $

Exercise 2 : Coefficients

Find the specified coefficient:

1. Coefficient of $ x^3 $ in $ (1 + x)^8 $
2. Coefficient of $ a^4b^2 $ in $ (2a - 3b)^6 $
3. Coefficient of $ x^{-2} $ in $ \left( x - \frac{1}{x} \right)^{10} $

Exercise 3 : Identities

Prove the following identities using the binomial theorem:

1. $ \sum_{k=0}^n \binom{n}{k} = 2^n $
2. $ \sum_{k=0}^n (-1)^k \binom{n}{k} = 0 $ for $ n \geq 1 $

Exercise 4 : General Term

1. Find the term independent of $ x $ in $ \left( x + \frac{1}{x^2} \right)^9 $.
2. Determine the middle term(s) in $ \left( 3x - \frac{y}{2} \right)^{10} $.

Exercise 5 : Approximations

1. Approximate $ (1.01)^5 $ using the binomial theorem.
2. Estimate $ (0.98)^6 $ to 4 decimal places.

Exercise 6 : Divisibility

Prove that for all $ n \in \mathbb{N} $:

1. $ 5^{2n+1} + 3^{n+2} \cdot 2^{n-1} $ is divisible by 19.
2. $ 4^n + 15n - 1 $ is divisible by 9.

Exercise 7 : Series Manipulation

1. Show that $ \sum_{k=1}^n k \binom{n}{k} = n \cdot 2^{n-1} $.
2. Evaluate $ \sum_{k=0}^n \binom{n}{k} \frac{1}{k+1} $.

Exercise 8 : Multinomials

1. Find the coefficient of $ x^3y^2z $ in $ (x + y + z)^6 $.
2. Expand $ (1 + x + x^2)^3 $ using multinomial coefficients.

Exercise 9 : Irrational Powers

For $ |x| < 1 $, use the generalized binomial theorem to:

1. Expand $ \sqrt{1 + x} $ up to $ x^3 $
2. Find the first 4 terms of $ (1 - x)^{-1/2} $

Exercise 10 : Combinatorial Proofs

1. Prove Vandermonde’s Identity:
   $$ \sum_{k=0}^r \binom{m}{k} \binom{n}{r-k} = \binom{m+n}{r} $$
2. Derive the identity:
   $$ \sum_{k=0}^n \binom{n}{k}^2 = \binom{2n}{n} $$

Exercise 11 : Non-Integer Exponents

Show that for $ |x| < 1 $ and $ \alpha \in \mathbb{R} $:

$$ (1 + x)^\alpha = \sum_{k=0}^\infty \binom{\alpha}{k} x^k $$

where $ \binom{\alpha}{k} = \frac{\alpha(\alpha-1)\cdots(\alpha-k+1)}{k!} $

Exercise 12 : Functional Equations

Find all functions $ f: \mathbb{R} \to \mathbb{R} $ satisfying:

$$ f(x+y) = \sum_{k=0}^n \binom{n}{k} f^{(k)}(x) y^k $$

where $ f^{(k)} $ denotes the $ k $-th derivative.

Exercise 13 : Asymptotics

Using Stirling’s approximation ($ n! \approx \sqrt{2\pi n} (n/e)^n $), estimate $ \binom{2n}{n} $ for large $ n $.

Exercise 14 : Number Theory

1. Prove that for a prime $ p $, $ \binom{p}{k} \equiv 0 \pmod{p} $ for $ 1 \leq k \leq p-1 $.
2. Show that $ \binom{2p}{p} \equiv 2 \pmod{p^2} $ for prime $ p \geq 5 $.

Exercise 15 : Generating Functions

1. Prove that the generating function for the sequence $ \binom{n}{0}, \binom{n}{1}, \dots, \binom{n}{n} $ is $ (1 + x)^n $.
2. Use generating functions to evaluate $ \sum_{k=0}^n \binom{n}{k}^3 $.

Exercise 16 : Inequalities

1. Show that for $ n \geq 1 $, $ \binom{2n}{n} \geq \frac{4^n}{2n+1} $.
2. Prove Chernoff’s bound: For $ 0 \leq p \leq 1 $ and $ X \sim \text{Binomial}(n, p) $,
   $$ P(X \geq (1+\delta)np) \leq \left( \frac{e^\delta}{(1+\delta)^{1+\delta}} \right)^{np} $$

Exercise 17 : Complex Analysis

Using contour integration, evaluate:

$$ \sum_{k=0}^n \binom{n}{k} \frac{(-1)^k}{k + 1/2} $$

Exercise 18 : Probability

Let $ X \sim \text{Binomial}(n, p) $. Compute $ E\left[ \frac{1}{1 + X} \right] $ exactly.