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Geometric Series Convergence and Divergence: A Complete Guide

Calculus: Master Limits, Derivatives & Integrals

Limits

Intermediate value theorem

Proving the Existence of Real Roots Using IVT

Intermediate Value Theorem

Application of Intermediate Value Theorem

Proving the Existence of a Real Root Using IVT

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Finding limits from graphs  

Continuity

Discontinuities of Rational Functions (denominator=0)

Finding limits algebraically - direct substitution 

Finding limits algebraically - when direct substitution is not possible

Infinite limits - vertical asymptotes  

Limits at infinity - horizontal asymptotes

Limit Evaluation

Limits at Infinity of Exponential Functions

Squeeze theorem

Quadratic Function with Integer Vertex

Limit laws

Evaluating Limits of Functions

limit laws

Evaluating Limits with specific limits given

Introduction to Calculus - Limits

Derivatives

Estimating derivatives from a table

Slope and equation of tangent line

Chain rule

Chain Rule

Differentiation of Trigonometric Function

Differentiate Radical Functions with Chain Rule

Differentiate: Trigonometric Functions

Chain rule: Differentiate Trigonometric Functions

Differentiation of Exponential Function

Differentiation of Exponential Functions

Differentiation: Chain Rule with Logarithmic Functions

Differentiation of Logarithmic Function

Product rule

Quotient rule

Differentiation using Quotient Rule

Differentiation using Chain Rule and Quotient Rule

Derivative of inverse trigonometric functions

Definition of derivative  

Definition of Derivative

Definition of derivative

Derivative using Definition

Vertical Tangent Line of a Function

Applications to definition of derivative

Power rule

Power Rule

Application of the Power Rule

Power rule application

Applying Power Rule

Power Rule with Negative Exponents

Power rule with rational exponents

Power rule and rational exponents

Derivative of trigonometric functions  

Derivative of Trigonometric Functions

Derivative of Composite Trig Functions

Derivative of exponential functions

Derivative of Exponential Functions

Differentiation of Composite Functions

Derivative of power and exponential functions

Derivative of logarithmic functions

Derivative of log with arbitrary base

Derivative of Natural Logarithm Functions

Derivative of Logarithmic Functions

Implicit differentiation

Higher order derivatives

Higher Order Derivatives

First and second derivatives

First and Second Derivatives

Second derivatives with implicit differentiation

Implicit Differentiation

Higher Order Derivatives with Repeating Patterns

Derivative Applications

Rolle&apos;s theorem

Quadratic approximation

Marginal profit, and maximizing profit & average profit

Elasticity of demand

Mean value theorem 

Mean Value Theorem

Mean value theorem

Mean Value Theorem Application

Critical number & maximum and minimum values

Critical Number & Maximum and Minimum Values

Linear approximation

Use of Linear Approximations in Estimating Square Roots

Linearization of Radical Functions

Linearization of Polynomial Functions

Linearization of Rational Functions

Linearization of Exponential Functions

Linearization of Logarithmic Functions

Linear Approximation of Trigonometric Functions

Optimization

Position velocity acceleration

Velocity of a Particle

Total Distance Traveled by a Particle

Comparative Analysis of Particle's Velocity and Acceleration

Curve sketching

Related rates

l'Hospital's rule

l'Hopital's rule

l'Hospital's Rule

Evaluating the limit using L'Hospital's Rule

l"Hospital"s rule

Demand, revenue, cost & profit

Marginal revenue, and maximizing revenue & average revenue

Marginal cost, and minimizing cost & average cost

Integrals

Antiderivatives

Derivatives and Antiderivatives

Derivative and Antiderivative of a Function

Riemann sum

Riemann Sum

Evaluating integrals with a Riemann sum

Evaluating integrals with a Riemann Sum

Riemann Sum with Trapezoid Approximation

Trapezoidal Approximation of Riemann Sums

Fundamental theorem of calculus

Fundamental Theorem of Calculus Part II

Fundamental Theorem of Calculus Part I

Fundamental Theorem of Calculus

Definite integral

Definite Integral

Integration Techniques

Nuerical integration

Numerical integration

Numerical Integration

U-Substitution

U-Substitution: Integrating Radical Functions

Not-So-Obvious U-Substitution

Trigonometric substitution

Trigonometric Substitution

Integration of rational functions by partial fractions

Improper integrals

Improper Integrals

Integration by parts

Integration using trigonometric identities

Integration Applications

Consumer and producer surplus

Continuous money flow

Areas between curves

Volumes of solids of revolution - Disc method

Volumes of solids with known cross-sections

Volumes of solid with known cross-sections

Volumes of solids of revolution - Shell method

Volumes of solid of revolution - Shell method

Arc length

Average value of a function

Order and solutions to differential equations

Separable equations

Separable Equations

Modeling with differential equations

Sequence and Series

Introduction to sequences

Monotonic and bounded sequences

Introduction to infinite series

Convergence and divergence of normal infinite series  

Convergence & divergence of geometric series 

Divergence of Geometric Series

Convergence & divergence of telescoping series 

math

<h2>What is calculus?</h2>
<p>Calculus math is the branch of mathematics involved with studying the rate of change, motion, and limits. As a joint discovery in Mathematics, Sir Isaac Newton and Gottfried Newbitz are credited as the Fathers of calculus for their early work in developing the fundamental concepts, theories and calculus rules we study today. Recent findings in the past decade however, suggest that this honour must now be split 4-fold to include Madhava and Nilakantha on the throne. Sir Isaac Newton was responsible for establishing the relationship between the two key topics in calculus: Differentials and Integrals. Gottfried Newbitz work was more focused on the rate of change in graphs alongside developing the mathematical notation for calculus that we use today.  Research has now confirmed that Madhava and Nilakantha were responsible for discovering infinite series in calculus, two centuries before Sir Isaac Newton and Gottfried Newbitz were born.</p>
<p>If you&apos;re just getting started with basic calculus or intro to calculus, differential calculus is the first order of business. Here you will learn how to calculate the slope of curves (e.g. parabolas), learn to use and apply the different derivative rules, and identify rates of change in velocity and acceleration.</p>
<p>In <a href="[[region_textbook.html|{"textbook_key":"calculus"}]]">Calculus 1</a>, <a href="[[region_textbook.html|{"textbook_key":"business-calculus"}]]">Business Calculus</a>, or any of your <a href="[[region_textbook.html|{"textbook_key":"ap-calculus-ab"}]]">AP Calculus classes</a>, you will also take a closer look at the concepts of limits and infinity. On the surface, these concepts appear to be at odds with one another. In the world of calculus however, this is far from true. Finding the limits of any graph requires you to look at what is happening at the microscopic level, dissecting the graph into infinitely small pieces and examining what happens as points on a graph get infinitesimally close together.  These foundational notions links differentials and integrals together.</p>
<p>Integral Calculus falls into the realm of advanced calculus. Topics covered here you&apos;ll likely see in <a href="[[region_textbook.html|{"textbook_key":"ap-calculus-ab"}]]">AP Calculus AB</a>, but more so in <a href="[[region_textbook.html|{"textbook_key":"ap-calculus-bc"}]]">AP Calculus BC</a> and <a href="[[region_textbook.html|{"textbook_key":"calculus2"}]]">Calculus 2</a>. Integral calculus is where you learn new methods of calculating area, volume and arc lengths of objects, learn integration techniques, and apply calculus to sequences and series. Science, Technology, Math, and Engineering (STEM) majors may be required to study <a href="[[region_textbook.html|{"textbook_key":"differential-equations"}]]">differential equations</a>, a higher level of specialized topics related to differential calculus. Here you will examine first and second order differential equations alongside Laplace transformations.</p>
<p>Learning calculus can be daunting, but don&apos;t feel intimidated by calculus - if you&apos;re at this stage, you have earned the right to be here and know that you&apos;re not going in completely unarmed. Calculus takes your knowledge of <a href="[[canon_textbook.html|{"textbook_key":"algebra-help"}]]">algebra</a>, <a href="[[canon_textbook.html|{"textbook_key":"trigonometry-help"}]]">trigonometry</a> and <a href="[[canon_textbook.html|{"textbook_key":"geometry-help"}]]">geometry</a> and elevates it to a new playing field where you will continue to expand and apply these skills in new ways. Let us help you <a href="[[signup.html]]">get started today</a>!</p>

<h2>Is calculus hard?</h2>
<p>Calculus has a notorious reputation for being difficult. Many who fear math have poised this perception of calculus. However, there are many who excel and enjoy calculus, some more so than the likes of <a href="[[canon_textbook.html|{"textbook_key":"algebra-help"}]]">Algebra</a> and <a href="[[canon_textbook.html|{"textbook_key":"trigonometry-help"}]]">Trigonometry</a>. The reality here however is that calculus is and can be hard - but it can be dealt with.</p>
<p>If you are just beginning with <a href="[[region_textbook.html|{"textbook_key":"pre-calculus"}]]">Precalculus for dummies</a> or even Calculus for dummies, there&apos;s no need to fret. Take your time to learn calculus essentials, familiarize yourself with calculus equations and follow along calculus examples to learn new techniques and problem solving strategies. Laying the groundwork here is important and will help you advance to more difficult calculus topics with ease. Remember that concepts are sequential and compounded on one another, so skip back if you feel like you need a quick refresher on common antiderivatives, more limits practice or a <a href="[[region_textbook.html|{"textbook_key":"calculus"}]]">complete Calculus 1 review</a>.</p>
<p>Like any other math topic, expect to face challenges. Furthermore, not all calculus problems are made equal. This is where we come in &ndash; getting the right calculus help will make a difference like night and day. Say goodbye to the repetitive monotony of calculus worksheets and join us for a real immersive, interactive and fun learning experience! At StudyPug we offer over 1000&apos;s of lessons, all brief yet comprehensive to help you tackle even the most arduous calculus questions.</p>
<p>Learn the lessons but don&apos;t stop there &ndash; journey with us as we dive deeper into the nitty gritty details, dissecting questions by going through step-by-step examples. Follow this up with easy practice to truly make the most out of your calculus exam prep. Our bank of 1179 questions is unrivalled by any, covering anything and everything under the sun! Let us supply you with all the practice and insight on the kinds of questions you can expect on your homework or upcoming test. Lastly, with our 24/7 help, never lose a night of sleep or throw a tantrum again. No more wasted time searching for a tutor, waiting for tutors to come online or commuting to a tutor &ndash; get calculus help now!</p>

<h2>How to learn calculus?</h2>
<p>When it comes to calculus, or any math subject, students are quick to ask "how to do calculus" instead of seeking "How to learn calculus".  In this modern day and age, to learn calculus online is quickly becoming the norm, whether it is for College Calculus or Calculus for beginners. With the advancement in technology and increase in accessibility of online resources, StudyPug aims to provide a superior service for those looking to learn calculus online, a calculus refresher, or access online calculus review material.</p>
<p>Learning calculus online begins with calculus videos. Unlike conventional math lessons, here at StudyPug, we&apos;ve condensed our lessons to be short, sweet, digestible and engaging. Give us 5-10 minutes of your time, and we&apos;ll clear up confusions, and lay down solid foundations. We&apos;ve also dedicated a large portion of our lessons to walk you through examples &ndash; learn the steps, how to avoid common pitfalls and master tricks and new techniques. Rest assured, our knowledgeable calculus tutors will help you lay your fears and qualms to rest. Still feeling lost? We&apos;ll take you back as far as <a href="[[region_textbook.html|{"textbook_key":"pre-calculus"}]]">Precalculus</a> or <a href="[[region_textbook.html|{"textbook_key":"algebra-help"}]]">Algebra</a> if necessary!</p>
<p>Looking for something more to chew on like our resident StudyPugs? Take a gander at our articles below each video lesson. These articles are your personal online notes where we break down calculus formulas, provide more calculus review, go over examples and frequently asked questions.</p>
<p>No learning is complete without actually testing your understanding. Go through as many calculus practice problems as you can to see how helpful our lessons were and how well you understood the concepts. Produce the confidence and sharpen the skills you need for that upcoming calculus test! If there is a calculus problem that escapes you, quickly backtrack to our walk-through examples or rewatch the lesson in no time.</p>

<h2>What textbooks do we use here at StudyPug?</h2>
<p>Choosing the best calculus textbook to use will be the least of your worries here at StudyPug. Our <a href="[[canon_textbook.html|{"textbook_key":"calculus-help"}]]">help page</a> covers calculus content found in textbooks like Calculus: Early Transcendentals, Springer Calculus II, Pearson Calculus, Calculus for dummies pdf, and many more! Keep calm and carry on with our Calculus help knowing that any calculus book or calculus textbook pdf you may be using will still be relevant and perhaps even more helpful once you&apos;ve used StudyPug! Never be put down by a mind-boggling calculus problem from your textbook again!</p>
<p>Feeling like you need more practice with differentiation calculus? Still stressed out with calculus integrals? Get into shape with our exercises and kick-start your success! With over 1000 calculus practice problems, be prepared for every style and level of calculus questions!</p>

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Mastering Geometric Series Convergence and Divergence

What You'll Learn

Identify when a geometric series converges or diverges based on the common ratio r

Apply the formula a/(1-r) to find the sum of convergent geometric series

Recognize geometric series in various algebraic forms and manipulate them to standard form

Determine the value of r and verify it falls between -1 and 1 for convergence

Show that geometric series with |r| 1 diverge to infinity

What You'll Practice

Algebraically manipulating series to reveal geometric form with factoring and exponent laws

Finding sums of convergent geometric series starting at n=0 or n=1

Converting series indices to match standard geometric series formulas

Proving divergence when the common ratio exceeds 1 in absolute value

Why This Matters

Before You Start — Make Sure You Can:

This Unit Includes

8 Video lessons

Practice exercises

Learning resources

Skills

Geometric Series

Convergence

Divergence

Infinite Series

Common Ratio

Series Summation

Algebraic Manipulation

Exponent Laws