Sequences and Series • Explore • Lenny.com

Lessons

Sigma Mastery

A comprehensive lesson on sigma notation where students master arithmetic, geometric, and infinite series through a hands-on rotation activity and collaborative problem-solving.

5

Factorial Fast Track

Students explore the rapid convergence of the infinite series for e, comparing the efficiency of factorials to the standard limit definition. The lesson bridges the gap between basic limits and the Taylor Series for e^x.

4

Euler's Constant Synthesis

An undergraduate-level exploration of Euler's number (e) that synthesizes its financial, analytical, and series-based definitions through a rigorous proof-based approach.

4

Golden Mean Convergence

Students will investigate the geometric mean property of the Fibonacci sequence, comparing estimations of high-index terms using geometric means versus the Golden Ratio method. The lesson explores the convergence of recursive sequences to geometric behavior for large n.

4

Golden Blueprint Discovery

Students will explore the mathematical connection between nature and geometry by calculating ratios of recursive sequences (Fibonacci and Lucas). Through this investigation, they will discover the Golden Ratio (Phi) and the concept of a mathematical limit.

3

Sigma Signal

A 12th-grade Pre-Calculus lesson focused on decoding, interpreting, and evaluating Sigma notation. Students transition from arithmetic series to the compact language of summation notation through video analysis and translation exercises.

3

Sequences

Convergence Lab Tests

A rigorous unit for 12th-grade Calculus students focusing on the Integral Test, p-series, and Comparison Tests (Direct and Limit) to determine the convergence of positive-term infinite series. Students will build a logical framework for selecting the most efficient convergence test for various mathematical structures.

13

Series Blueprinting

This advanced sequence introduces powerful tools for analyzing series with factorials and powers, leading to the concept of power series. Students master the Ratio and Root tests, explore absolute versus conditional convergence, and conclude by connecting series to functions through Taylor polynomials.

21

Rigorous Proofs in Analysis

An undergraduate-level introduction to Real Analysis focusing on the formal epsilon-N definition of limits, proof construction, Cauchy sequences, and the Bolzano-Weierstrass Theorem. Students transition from computational calculus to rigorous mathematical proof.

10

Iterative Insights

This sequence explores numerical analysis through the lens of sequences, focusing on iterative methods to approximate solutions to complex equations. Students investigate fixed-point iteration, Newton's method, convergence rates, and the transition into chaotic behavior.

10

Infinite Destinations

This sequence guides undergraduate students from an intuitive understanding of sequence limits to rigorous analysis using algebraic laws, the Squeeze Theorem, L'Hôpital's Rule, and the Monotone Convergence Theorem. Students will explore how infinite processes behave as they approach infinity, bridging the gap between discrete sequences and continuous calculus.

11

System Dynamics and Chaos

A graduate-level exploration of discrete dynamical systems, moving from linear growth models to the complex, chaotic behavior of the logistic map. Students apply recursive sequences to model biological and economic phenomena, emphasizing stability analysis and bifurcation theory.

10

Videos

Mastering Math with the Desmos Online Graphing Calculator

This comprehensive tutorial introduces students and educators to the Desmos online graphing calculator, a powerful and free tool for visualizing mathematics. The video begins with the basics of accessing the site and graphing simple functions like lines and parabolas, then progressively moves to more advanced topics including trigonometry, polar coordinates, and calculus operations. The narrator demonstrates not just how to plot graphs, but how to use the tool to solve equations, factor quadratics, and visualize systems of inequalities. Key mathematical themes include function families (linear, quadratic, cubic, absolute value, exponential, logarithmic), systems of equations and inequalities, trigonometry (degrees vs. radians), and statistical analysis (mean, standard deviation). The video also highlights the calculator's computational abilities, such as evaluating definite integrals, summations, and permutations, making it a versatile utility for various levels of math. For the classroom, this video serves as an excellent training resource for students transitioning from handheld calculators to computer-based tools. It allows teachers to show students how to verify their algebraic work visually, such as checking factors by looking at x-intercepts or solving systems by finding intersection points. The visual nature of the tool helps bridge the gap between abstract algebraic manipulation and concrete graphical understanding.

The Organic Chemistry Tutor

13mins 58s

What Is the Number e and How Do We Calculate It?

This educational video provides a comprehensive overview of the mathematical constant e, also known as Euler's number. It begins by defining e as an irrational number approximately equal to 2.718, commonly found in exponential functions and logarithms. The narrator then explores the origins of e through three distinct mathematical approaches: the formula for compound interest as the compounding frequency approaches infinity, the limit of (1 + n)^(1/n) as n approaches zero, and the infinite series of the reciprocals of factorials. Key themes include the relationship between discrete and continuous growth, the concept of limits and infinity in mathematics, and the unique properties of exponential functions in calculus. The video bridges the gap between algebraic financial formulas and advanced calculus concepts, demonstrating how a simple question about interest rates leads to the discovery of a fundamental constant of the universe. For educators, this video serves as an excellent bridge between Algebra II, Pre-Calculus, and Calculus. It visualizes abstract definitions, provides concrete numerical evidence for limits, and introduces the profound calculus properties of e^x—specifically that its slope and the area under its curve are equal to its value. This makes it a versatile tool for introducing the constant to beginners or deepening the understanding of advanced students studying derivatives and integrals.

The Organic Chemistry Tutor

12mins 11s