Course summary: ... ---- # 1. Power Series I | Definitions | Theorems | Examples | | ---------------------------------------------------------- | --------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------ | | [[Power Series]]. | | | | [[Radius of Convergence of Complex Power Series]]. | [[Radius of Convergence of Complex Power Series]]. | [[Radius of Convergence of Geometric Series]]. | | | [[Radius of Convergence of Absolute Real Power Series About Zero]]. | [[Radius of Convergence of Log Series]]. | | | | [[Throwing away terms from Geometric Series Maintains Its Radius of Convergence]]. | | | | Be able to use [[Ratio Test for Series]] to find radius of convergence. | | | *[[Continuity of Real Power Series About Zero on Interval of Convergence]] | | | [[Real Exponential Function as Power Series]]. | [[Real Exponential Function of Sum]]. | *[[Cauchy Product of Univariate Real Power Series About Zero Converges to Product]]. | | | *[[Lower Bound for Real Exponential Function]]. | *[[Equivalence of Euler's Number as Convergent Real Sequence and Real Power Series]]. | | | *[[Upper Bound for Real Exponential Function for all x<1]]. | | | | [[Real Exponential Function is Strictly Increasing]]. | | | [[Real Natural Logarithm Function]]. | [[Inverse of Real Exponential Function Exists, is Strictly Increasing and Continuous]]. | | | | [[Real Natural Logarithm of Product]]. | | | [[Real Power of Real Number]]. | [[Real Power Extends Integer Power of Real Number]]. | | | | [[Product of Real Powers of a Real Number]]. | | | | [[Real Natural Logarithm of Real Power of Real Number]]. | | | | [[Real Power of Real Power of Real Number]]. | | | | [[Real Exponential Function is Real Power of Euler's Number]]. | | | | [[Tangent to Real Natural Logarithm at x = 1]]. | [[AM-GM Inequality]]. | # 2. Limits and The Derivative | Definitions | Theorems | Examples | | ---------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------- | -------------------------------------------------- | | [[Limit of Real Function at a Point]]. | *[[Limit of Continuous Real Function at a Point]]. | [[Dini's Theorem]]. | | | [[Limit of Real Function by Convergent Real Sequences]]. | | | | [[Algebra of Limits of Real Functions]]. | | | One sided limits: [[Left Limit of Real Function at a Point]]; [[Right Limit of Real Function at a Point]]. | | | | Infinite limits: [[Properly Divergent Univariate Real Function]]. | | | | [[Limit at Infinity of Univariate Real Function]]. | | | | [[Fréchet Differentiation]]. | [[Differentiablity implies Continuity]]. | | | [[Derivative of Real Function]]. | [[Derivative of Sum of Differentiable Real Functions]]. | | | | [[Derivative of Product of Differentiable Real Functions]]. | | | | [[Derivative of Monomials]]. | | | | *[[Weirestrass-Caratheodory Criterion for Differentiability of Real Function]]. | | | | *[[Chain rule for derivative]]. | | | | | | | | [[Rolle's Theorem]]. | | | | *[[Mean value theorem]]. | | | | [[Positive Derivative Implies Strictly Increasing Real Function]]. | | | | [[Real Function with Zero Derivative is Constant]]. | f'(x)=f(x) sol | | | | | | | [[Derivative of Inverse of Strictly Monotonic Differentiable Real Function]]. | [[Derivative of Real Natural Logarithm Function]]. | | | | [[Derivative of Arcsine]]. | # 3. Power Series II | Definitions | Theorems | Examples | | -------------------------------- | --------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------- | | | [[Radius of Convergence of Derivative of Real Power Series About Zero]]. | | | | *[[Power Series is Termwise Differentiable within Radius of Convergence]]. | Show that $f(x)=\sum_{n=1}^{\infty}x^{n}/n^{2}$ satisfies the DE: $xf''(x)+f'(x)=\frac{1}{1-x}$ | | | [[Derivative of Real Exponential Function]]. | Find power series solution $xy''+y'+xy=0$ | | | [[Derivative of Real Natural Logarithm Function]]. | | | [[Cosine as Real Power Series]]. | [[Cosine of Sum]]; [[Sine of Sum]]. | | | [[Sine as Real Power Series]]. | [[Pythagorean Trigonometric Identity]]; [[Unit Speed Parametrisation of Unit Circle]]. | | | | [[Exponential Function in Terms of Trigonometric Functions (Euler's Formula)]]. Proof not given. Should be able to use to prove addition formulae. | | | [[Tangent Function]]. | | | # 4. Taylor's Theorem | Definitions | Theorems | Examples | | ----------- | ---------------------------------------------------------------------------- | -------------------------------------------------------- | | | [[Cauchy Mean Value Theorem]]. | | | | [[L'Hôpital's Rule For Limit of Quotient at a Point]]. | | | | [[L'Hôpital's Rule For Limit of Quotient at Infinity]]. | | | | | | | | [[Taylor's Theorem With Lagrange Remainder for Real Function]]. | [[Taylor Expansion of Real Natural Logarithm Function]]. | | | [[Taylor's Theorem With Schlömilch Remainder for Univariate Real Function]]. | [[Binomial Series]]. | | | [[Taylor's Theorem With Cauchy Remainder for Univariate Real Function]]. | | # 5. Riemann Integral Note that the in this section, the Darboux integral is called the Riemann integral which is acceptable as their definitions can be proven to be equivalent. | Definitions | Theorems | Examples | | ----------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------- | | [[Finite Partition of Closed Real Interval]]. | | | | [[Riemann integration]]; [[Riemann integration]]. | [[Upper Riemann Sum is Never Smaller than Lower Riemann Sum for The Same Finite Partition]]. | | | [[Riemann integration]]; [[Riemann integration]]. | | | | [[Riemann integration]]. | | Show that $f(x)=\begin{cases}1 & x\in \mathbb{Q} \\0 & x \not\in\mathbb{Q} \end{cases}$is not integrable; | | | | *[[Darboux Integral of Thomae's Function is Zero]] | | | | | | [[Refinement of Finite Partition of Closed Real Interval]]; [[Common Refinement of Finite Partition of Closed Real Interval]]. | *[[Upper & Lower Riemann Sums of Refinement]]. | [[Darboux Integral of Constant]]. | | | [[Upper & Lower Riemann Sums of Refinement]]. | | | | [[Upper Riemann Integral is Never Smaller than Lower Riemann Integral]]. | | | | [[Riemann's criterion for integrability]]. | | | | | | | [[Uniformly Continuous Real Function]]. | [[Continuous Real Function on Closed Real Interval is Uniformly Continuous]]. | [[Uniformly Continuous Real Function on Bounded Real Interval is Bounded]]. | | | [[Continuous real functions are Riemann Integrable]]. | [[Existence of Limits of Uniformly Continuous Real Function on Open Real interval at End-points]] | | | [[Monotone Real Function is Darboux Integrable]]. | *[[Piecewise Continuous Real Function is Darboux Integrable]]. | | [[Mesh Size of Finite Partition of Closed Real Interval]]. | *[[Upper Darboux Sums of Darboux Integrable Function Converge to Darboux Integral as Mesh Size Approaches Zero]]. | | | | | | | [[Riemann integration]]. | Properties of Riemann/Darboux Integral: | | | | *[[Linearity of Riemann integration]] - proved using [[Triangle Inequality for Supremum of Function on Interval]]. | | | | [[Monotonicity of Riemann Integral]] - was left as exercise. | | | | *[[Continuous Real Function of Darboux Integrable Function is Darboux Integrable]]. | If $f$ is integrable then $f^{2}$ is integrable on the same closed real interval. | | | [[Product of Riemann integrable functions is also]]. | [[Cauchy-Schwartz Inequality for Darboux Integral]]. | | | [[Triangle Inequality for Riemann Integral]]. | | | | [[Finite Additivity of Riemann integral]] - sketch of proof given. | | | | | | | | [[Fundamental theorem of calculus]]. | | | | [[Fundamental theorem of calculus]]. | | | | [[Integration by Parts]]. | | | | [[Integration by Substitution]]. | | | | | | | [[Improper Riemann Integral]]; [[Improper Integral Over Half Open Interval]]; [[Improper Integral Over Unbounded Closed Interval]]. | *[[Comparison Test for Improper Integrals over Unbounded Closed Interval]]. | [[Integral of Monomials]]. | | | | | | [[Integral Form of Gamma Function]]. | *[[Equivalence of Integral and Euler Forms of Gamma Function]]. | [[Gamma Function Extends Factorial]]. | | [[Euler Form of Gamma Function]]. | | [[Stirling's Approximation]]. | # Appendix. The Radius of Convergence Formula | Definitions | Theorems | Examples | | ----------- | --------------------------------------------------------------------------------- | -------- | | | [[Formula for Radius of Convergence of Univariate Real Power Series About Zero]]. | |