1. **Unit I: Sequences** Definition: A sequence is an ordered list of numbers. It can be finite (ends after a certain number of terms) or infinite (goes on forever). - Finite sequence example: $1, 2, 3, 4, 5$ - Infinite sequence example: $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots$ Bounded sequence: A sequence where all terms lie within some fixed interval. Unbounded sequence: Terms grow without bound. Limit of a sequence: The value the terms approach as $n \to \infty$. - Convergent sequence: Has a finite limit. - Divergent sequence: Does not have a finite limit. - Oscillatory sequence: Terms oscillate without settling. Monotonic sequence: Always increasing or always decreasing. Important theorem: Every convergent sequence is bounded. Operations on convergent sequences: - Sum, difference, product, and quotient (if denominator nonzero) of convergent sequences are convergent. Example: Find limit of $a_n = \frac{1}{n}$. - As $n \to \infty$, $a_n \to 0$. So, sequence converges to 0. 2. **Unit II: Infinite Series** Definition: Sum of terms of a sequence. - Positive term series: All terms positive. - Geometric series: $S = a + ar + ar^2 + \dots$ with ratio $r$. - $p$-series: $\sum \frac{1}{n^p}$. Tests for convergence: - Comparison test - D’Alembert’s ratio test - Raabe’s test - Cauchy’s root test Alternating series: Terms alternate in sign. Leibnitz’s theorem: Conditions for convergence of alternating series. Examples: - Sum geometric series with $|r|<1$: $$S = \frac{a}{1-r}$$ - Exponential series: $e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$ 3. **Unit III: Differential Calculus** Differentiability: Function has a derivative at a point. - Every differentiable function is continuous, but not vice versa. Theorems: - Intermediate Value Theorem - Rolle’s Theorem - Lagrange’s Mean Value Theorem - Cauchy’s Mean Value Theorem Taylor’s theorem: Approximate functions by polynomials. Maclaurin’s series: Taylor series at 0. Indeterminate forms: Use L’Hospital’s rule to evaluate limits like $\frac{0}{0}$ or $\frac{\infty}{\infty}$. Example: Find derivative of $f(x) = x^2$. - $f'(x) = 2x$ 4. **Unit IV: Functions of Two and Three Variables** Concepts: - Limit and continuity for multivariable functions. - Partial derivatives: Derivative with respect to one variable keeping others constant. - Homogeneous functions: Satisfy $f(tx, ty) = t^n f(x,y)$. - Euler’s theorem for homogeneous functions. - Total derivatives: Combine partial derivatives. - Jacobian: Determinant of partial derivatives matrix. - Maxima and minima for functions of two variables. Example: Find partial derivatives of $f(x,y) = x^2 y + y^3$. - $\frac{\partial f}{\partial x} = 2xy$ - $\frac{\partial f}{\partial y} = x^2 + 3y^2$