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Mastering Monotonic and Bounded Sequences in Mathematics

Integral Calculus: Master Advanced Integration Techniques

Integrals

Antiderivatives

Derivatives and Antiderivatives

Derivative and Antiderivative of a Function

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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

Parametric Equations and Polar Coordinates

Defining curves with parametric equations

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

Parent Assignments

Challenge Zone

Foundation Review

Integral calculus made easy!
Get better marks in calculus class with our complete Integral Calculus help. Our calculus tutors cover all topics you will see in any typical Integration class that deals with single variable functions.

Keeping with your textbook and class, our Integral calculus video lessons walk you through all topics in integral calculus such as, Antiderivative, Integral of trig functions, Definite integral, Indefinite integral, Integration by substitution, Improper integrals and more. Learn the concepts with our tutorials that show you step-by-step solutions to even the hardest integral calculus questions. Then, reinforce your understanding with tons of integral calculus practice.
All our lessons are taught by experienced calculus teachers. Let&apos;s finish your homework in no time, and ACE that exam.

Integral Calculus

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Intro

Ex.1

Ex.2

Ex.3

Intro to monotonic sequences: increasing, decreasing, and neither

Understanding bounded sequences and the monotonic convergence theorem

Showing n squared is a monotonic increasing sequence

Showing 1 over 3^n is a decreasing and monotonic sequence

Showing n/(n+1) is increasing despite a larger denominator

Identifying monotonic behavior in a sequence given by its terms

Determining if (-1)^n / n² is bounded above, below, or both

Determining if the sequence n cubed is bounded below, above, or both

Determining if an alternating sequence is bounded above, below, or neither

Determining if a_n = -n^4 is bounded above, below, or both

Proving convergence of 3/n³ using monotonic and bounded test

Constant sequence: bounded but not monotonic, testing Theorem 7

Showing Theorem 7 cannot determine convergence of sqrt(n)

In this section, we will be talking about monotonic and bounded sequences. We will learn that monotonic sequences are sequences which constantly increase or constantly decrease. We also learn that a sequence is bounded above if the sequence has a maximum value, and is bounded below if the sequence has a minimum value. Of course, sequences can be both bounded above and below. Lastly, we will take a look at applying theorem 7, which will help us determine if the sequence is convergent. One important to note from the theorem is that even if theorem 7 does not apply to the sequence, there is a possibility that the sequence is convergent. It's just that the theorem will not be able to show it.

What are monotonic and bounded sequences?

Understanding Monotonic and Bounded Sequences

What You'll Learn

Identify increasing and decreasing sequences by comparing consecutive terms

Recognize monotonic sequences as those that are strictly increasing or strictly decreasing

Determine if sequences are bounded above, bounded below, or both

Apply Theorem 7 to prove convergence when sequences are both monotonic and bounded

Analyze sequence behavior by plotting terms and examining patterns

What You'll Practice

Testing sequences with formulas like n², 1/3, and n/(n+1) for monotonic behavior

Graphing sequence terms to identify bounds and determine convergence

Evaluating whether sequences with alternating signs are bounded above or below

Applying Theorem 7 to verify convergence using monotonicity and boundedness

Why This Matters

Before You Start — Make Sure You Can:

This Unit Includes

13 Video lessons

Practice exercises

Learning resources

Skills

Monotonic Sequences

Bounded Sequences

Convergence

Theorem 7

Sequence Analysis

Increasing Sequences

Decreasing Sequences