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Geometric Series Convergence and Divergence: A Complete Guide

Calculus 2: Master Advanced Integration & Series

Parametric Equations and Polar Coordinates

Defining curves with parametric equations

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Tangent and concavity of parametric equations

Area of parametric equations

Arc length and surface area of parametric equations

Arc Length of a Parametric Curve

Polar coordinates

Tangents of polar curves

Area of polar curves

Arc length of polar curves

Integrals

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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

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

Areas between curves

Volumes of solids of revolution - Disc method

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

Introduction to differential equations

Order and solutions to differential equations

Separable equations

Separable Equations

Applications to differential equations

Modeling with differential equations

Slope fields

Understanding Slope Fields

Euler's method

Euler's Method

Euler"s Method

Euler"s method

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

Divergence of harmonic series

Divergence of Harmonic Series

P Series

Convergence and Divergence of P Series

Alternating series test

Alternating Series Test

Convergence of the Alternating Series Test

Divergence test

Divergence Test

Comparison & limit comparison test

Comparison & Limit Comparison Test

Integral test

Integral Test for Series Convergence

Convergence/Divergence of Integral Test

Ratio test

Ratio Test

Convergence & Divergence of Ratio Test

Root test

Root Test

Root Test for Series Convergence

Convergence & Divergence of Root test

Absolute & conditional convergence

Absolute & Conditional Convergence

Radius and interval of convergence with power series

Radius of Convergence of Logarithmic Function

Radius and Interval of Convergence with Power Series

Radius of Convergence of Tangent Function

Functions expressed as power series

Taylor series and Maclaurin series

Approximating functions with Taylor polynomials and error bounds

Parent Assignments

Challenge Zone

Foundation Review

<p>Our calculus tutors have you covered with our complete help for any calculus 2 courses that involve single-variable functions, whether it is Business Calculus 2, or <a href="http://www.ibo.org/globalassets/publications/recognition/5_furthermathhl.pdf">IB Mathematics HL</a>. Learn calculus 2 with ease!</p>
<p>Our comprehensive lessons on calculus 2 cover help on topics like Riemann Sum, Trig substitution, Alternating series test, Integral of trig functions, P series test, Integration by partial fractions and so much more. Learn the concepts with our online calculus 2 tutorials that show you step-by-step solutions to even the hardest integration calculus problems. Then, strengthen your understanding with tons of integration practice. </p>
<p>All our lessons are taught by experienced calculus 2 teachers. Let&apos;s finish your homework in no time, and ACE that final.</p>

Calculus 2

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math

<h2>What is Calculus 2?</h2>
<p>When it comes to learning Calculus 2, you&apos;ll need to come in with a solid understanding of the fundamentals of <a href='[[region_textbook.html|{"textbook_key":"calculus"}]]'>Calculus I</a> as you won&apos;t go far in this second course without knowing the ins-and-outs of limits, derivatives and applications of derivative. Our Calculus 2 course outline is organized logically with comprehensive notes each step of the way.</p>
<p>For this course, our certified StudyPug Calculus 2 tutors will guide you through six topics:</p>
<ul type="dash">
<li>Integrals</li>
<li>Integration Techniques</li>
<li>Integration Applications</li>
<li>Differential Equations</li>
<li>Sequence and Series</li>
<li>Parametric Equations and Polar Coordinates.</li>
</ul>

<h2>Is Calculus 2 Hard?</h2>
<p>Unless you are majoring in math, Calculus 2 will likely be the most difficult math course you will take thus far, mainly because it requires you to have a fair amount of maturity and creativity with math that you may not have needed up to this point. The hardship in understanding Calculus 2 isn&apos;t necessarily from committing the numerous formulas to memory, but instead, it can be difficult for knowing when to employ those Calculus 2 formulas. Sure, you&apos;re going to need to memorize all the fomulas, but you&apos;ll need to know the nuances of how the formulas operate, which in turn, helps you determine which Calculus 2 formula to use. It can be a dizzying amount of mental work that can be overwhelming for a student.</p>
<p>The perception of Calculus 2 being so difficult comes from the fact that a lot of students are actually surprisingly well prepared for Calculus 1 whereas this course often feels like a rude awakening. Compared to Calculus 1, it's relatively difficult to think of "real world application" that specifically depend on Calculus 2 concepts, and so there is a wide chasm between the material and students being able to relate to it. An easy Calculus 2 test can quickly turn into an impossible one when you start breaking down a nasty-looking integral that really only required a simple trig substitution.</p>
<p>Luckily, our Calculus 2 course is designed in a way where the complicated subject is broken into manageable chunks where we take time to walk you through those intricate nuances to bolster your grasp of the topic. If you&apos;re looking for Calculus 2 for dummies, we&apos;ve got a smart solution that will turn you into a Calculus 2 master in no time.</p>

<h2>What comes after Calculus 2?</h2>
<p>Firstly, it is highly recommended that you go through all of Calculus 1 before taking this Calculus 2. There&apos;s no real way around that as jumping into this course without a sound working knowledge of Calculus 1 is ill-advised. When anyone teaches Calculus 2, they will have to build on terms and formulas taught in Calculus 1. Besides that, another prerequisite for this course can be either <a href='[[region_textbook.html|{"textbook_key":"business-calculus"}]]'>Business Calculus</a> or <a href='[[region_textbook.html|{"textbook_key":"ap-calculus-ab"}]]'>AP Calculus AB</a> too.</p>
<p>After StudyPug has helped you conquer Calculus 2, your follow-up course should be either Calculus 3 or <a href='[[region_textbook.html|{"textbook_key":"linear-algebra"}]]'>Linear Algebra</a> where our instructors will further build on the knowledgebase you&apos;ve gained from taking this course.</p>

<h2>Who needs to take Calculus 2?</h2>
<p>In university, Calculus 2 is a required course for several academic disciplines including most science and math related majors like Physics, Engineering, Computer Science, etc. Basically, if you are in a field that uses math or science, you will absolutely encounter Calculus 2 in your course requirements.</p>
<p>Our Calculus 2 lessons are compatible with Calculus 2 textbooks such as  <a href="https://www.goodreads.com/book/show/61298.Calculus">Calculus: Early Transcendentals</a>; <a href="http://www.springer.com/us/book/9780387909752">Springer Calculus II</a>, <a href="https://www.pearson.com/us/higher-education/math---science/mathematics/calculus.html">Pearson Calculus</a>; and more.</p>

<h2>How to study for Calculus 2?</h2>
<p>In StudyPug, our education philosophy revolves around explaining complicated topics like Calculus 2 in an approachable manner. Instead of trying to parse truths from Googling out specific topics you may have a missed a class on, you can come here at any time to learn from our Calculus 2 tutorials to catch up. When searching for Calculus 2 topics, be specific so we can be precise in helping out with that burning Calculus 2 question that&apos;s been on your mind.</p>

<h2>Here are just some of the reasons why you&apos;ll want to get Calculus 2 help with StudyPug:</h2>
<ul type="dash">
<li>Our Calculus 2 tutorials are comprehensive, which means that we can answer any Calculus 2 questions that you may have. Whether it be Riemann Sum, Trig substitution, Alternating series test, Integral of trig functions, or P series test, StudyPug will provide answers you seek.</li>
<li>Our Calculus 2 practice is available immediately after each Calculus 2 lesson for you to practise the lessons&apos; material. After all, you can&apos;t master Calculus 2 without flexing that knowledge through practice.</li>
<li>Our Calculus 2 tutorials are always available to help you for all your Calculus 2 needs. Even if you have to miss school from sickness or a vacation, you can always access these online Calculus 2 lessons which include an intro to the topic, practice, and a video tutorial with StudyPug 2 tutors.</li>
</ul>

<h2>What do you learn in Calculus 2?</h2>
<p>Calculus 2 is a bit of a mixed assortment of mathematical topics. The beginning of Calculus 2 deals with building on those aforementioned fundamentals of derivatives you should have touched on in Calculus 1.</p>
<p>You&apos;ll spend your early time formalizing antiderivatives, Riemann sums, the fundamental theorem, etc.</p>
<p>After that, you'll jump into integration techniques, which is usually the most difficult part of the course for most students. Here, you will pick up several different tricks to solve integration problems, but you&apos;ll also learn that integration is not as straightforward as differentiation.</p>
<p>From there, you'll then proceed to applications of integration to solve some very "fun" mathematical problems, like the area between two curves, volumes of different types of solids, average value of function, arc length, etc.</p>
<p>Next, you'll study differential equations where you will be exposed to an overview of the topic with its various solutions.</p>
<p>Following that, Sequence and Series is another heavy topic, but we divide it into 19 sub-topics so you can learn Calculus 2 at your own pace. Everything is covered here from an introduction to sequences to approximating functions with Taylor polynomials and error bounds.</p>
<p>Finally, we close out this course with a dive into Parametric Equations and Polar Coordinates. In this section, we teach you everything from how to define curves with parametric equations to understanding the arc length of polar curves.</p>
<p>If this sounds terrifying, fret not. StudyPug is committed to give you dedicated help with Calculus 2 questions by offering useful and easy-to-absorb Calculus 2 lessons.</p>

<h2>Using StudyPug for Calculus 2</h2>
<p>Stuck on a whether to use u-substitution or numerical integration? Need to quickly look up which formula to use? Need an instant refresher on how to calculate the area between two curves? Well StudyPug can help you with any Calculus 2 problem as we offer thorough Calculus 2 homework help.</p>
<ul type="dash">
<li>To find the Calculus 2 topic where your Calculus 2 question belongs to, you can either use our search system to find the correct Calculus 2 topic, or just browse through the large list of Calculus 2 topics that we cover.</li>
<li>Watch a step-by-step Calculus 2 tutorial within each topic to see how our Calculus 2 teacher solves the question along with his best Calculus 2 tips and tricks!</li>
<li>If you are rusty on the Calculus 2 concept as a whole, it&apos;s always recommended to start by watching introduction to the topic for your Calculus 2 homework.</li>
<li>Again, StudyPug serves up quality Calculus 2 homework help available for you around the clock.</li>
</ul>

https://dmn92m25mtw4z.cloudfront.net/img/textbook/textbook-calculus2.png

Intro

Intro to geometric series and the convergence formula

Ex.1

Ex.2

Algebraically manipulating a series to apply the geometric series formula

Splitting a complex series into three geometric series to find the sum

Manipulating exponents to apply the geometric series formula

Evaluating a geometric series with exponent manipulation

Showing a geometric series with r = 3/2 is divergent

Showing divergence of a geometric series with r greater than 1

Splitting a series to determine divergence when one part diverges

In this section, we will take a look at the convergence and divergence of geometric series. We've learned about geometric sequences in high school, but in this lesson we will formally introduce it as a series and determine if the series is divergent or convergent. For the first few questions we will determine the convergence of the series, and then find the sum. For the last few questions, we will determine the divergence of the geometric series, and show that the sum of the series is infinity.

Convergence & divergence of geometric series 

Convergence and divergence of geometric series 

Convergence and divergence in a geometric series

Convergence and divergence of normal infinite series  

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