This first, longer part of the guide concerns the behavior of numerical series and whether they have an actual sum as the number of terms approaches infinity. There are many tests and methods that we can use to answer this question, each working on specific cases. However, before we can answer this, we need to first talk about what sequences and series are first.
Before we start talking about series, we need to talk about sequences and some terminology about sequences.
A sequence is just a list of terms related by a common pattern to each other.
Here is how we represent sequences:
Like functions, sequences have limits! These are found in much the same way that the limit of a function is found, but in this unit, we are only interested in finding out the limit as n approaches ♾️. Also note that all limit properties that hold for regular functions hold for sequences as well.
There is also a special theorem that holds for sequence limits as well which will be useful for.
We will not be doing any examples specifically geared towards finding limits of sequences, but these will be used in other applications as well.
The first of these is finding out whether a sequence is convergent or divergent. The words convergent and divergent will show up a lot in this unit, so stay alert!
Before we move on to series, there is some terminology that we have to cover real quick when we talk about sequences.
These definitions lead to a theorem about sequence convergence.
Now that we understand sequences, let’s start talking about series!
Like sequences, series can also converge or diverge. We will list their definitions below.
Since the series we just did has a finite value for the infinite partial sum, the series converges. In the rest of the first part of the unit, we will find a way to determine whether a series is convergent or divergent, so don’t worry if you don’t know this yet!
Here are some properties of convergent series that will be helpful throughout the unit!
Define the following terms.
5. Monotonic Sequence:
6. Bounded Sequence:
7. Convergent/Divergent Sequence:
8. Convergent/Divergent Series:
✍️ Free Response Questions (FRQ)
Calculus Free Response Questions
👑 Unit 1: Limits & Continuity
1.5Determining Limits Using Algebraic Properties of Limits
1.6Determining Limits Using Algebraic Manipulation
1.10Exploring Types of Discontinuities
1.11Defining Continuity at a Point
1.12Confirming Continuity over an Interval
🤓 Unit 2: Differentiation: Definition & Fundamental Properties
2.4Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
🤙🏽 Unit 3: Differentiation: Composite, Implicit & Inverse Functions
3.0Unit 3 Overview: Differentiation: Composite, Implicit, and Inverse Functions
3.1The Chain Rule
3.3Differentiating Inverse Functions
3.4Differentiating Inverse Trigonometric Functions
👀 Unit 4: Contextual Applications of the Differentiation
4.2Straight-Line Motion: Connecting Position, Velocity, and Acceleration
4.4Intro to Related Rates
4.6Approximating Values of a Function Using Local Linearity and Linearization
✨ Unit 5: Analytical Applications of Differentiation
5.0Unit 5 Overview: Analytical Applications of Differentiation
5.2Extreme Value Theorem, Global vs Local Extrema, and Critical Points
5.3Determining Intervals on Which a Function is Increasing or Decreasing
5.4Using the First Derivative Test to Determine Relative (Local) Extrema
5.7Using the Second Derivative Test to Determine Extrema
🔥 Unit 6: Integration and Accumulation of Change
6.11Integrating Using Integration by Parts
💎 Unit 7: Differential Equations
7.0Unit 7 Overview: Differential Equations
7.7Finding Particular Solutions Using Initial Conditions and Separation of Variables
🐶 Unit 8: Applications of Integration
8.1Finding the Average Value of a Function on an Interval
8.2Connecting Position, Velocity, and Acceleration of Functions Using Integrals
8.3Using Accumulation Functions and Definite Integrals in Applied Contexts
8.4Finding the Area Between Curves Expressed as Functions of x
8.5Finding the Area Between Curves Expressed as Functions of y
8.6Finding the Area Between Curves That Intersect at More Than Two Points
8.7Volumes with Cross Sections: Squares and Rectangles
8.8Volumes with Cross Sections: Triangles and Semicircles
8.9Volume with Disc Method: Revolving Around the x- or y-Axis
8.10Volume with Disc Method: Revolving Around Other Axes
8.11Volume with Washer Method: Revolving Around the x- or y-Axis
🦖 Unit 9: Parametric Equations, Polar Coordinates & Vector Valued Functions (BC Only)
9.0Unit 9 Overview: Parametric Equations, Polar Coordinates, and Vector-Valued Functions
9.1Defining and Differentiating Parametric Equations
♾ Unit 10: Infinite Sequences and Series (BC Only)
10.0Unit 10 Overview: Infinite Series and Sequences
10.1Defining Convergent and Divergent Infinite Series
10.6Comparison Tests for Convergence
10.7Alternating Series Test for Convergence
10.11Finding Taylor Polynomial Approximations of Functions
10.14Finding Taylor or Maclaurin Series for a Function
🧐 Multiple Choice Questions (MCQ)
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