Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Algebra → Sequences and Series ↓


Convergence and Divergence


In mathematics, sequences and series play a vital role in understanding many mathematical concepts. For class 11 students, it is essential to be familiar with the terms “convergence” and “divergence”. These concepts help us understand the behaviour of sequences and series as they proceed indefinitely. Let’s delve deeper into these terms, exploring them with both graphical and textual examples.

Sequences and series: An introduction

Before we can fully understand convergence and divergence, it is important to understand what a sequence and a series are. A sequence is simply a list of numbers in a specific order. For example, 1, 2, 3, 4, 5, ... is a sequence where each number is one more than the previous number.

On the other hand, a series is what you get when you add numbers in a sequence. For example, if we take the sequence 1, 2, 3, 4, 5, ..., the series would be written as 1 + 2 + 3 + 4 + 5 + ...

What is convergence?

A sequence is said to be convergent when the numbers get closer to a specific value, called the "limit," as they move along the sequence. For example, consider the sequence:

1, 0.5, 0.25, 0.125, 0.0625, ...

If we observe, every term in the sequence is half of the previous term. As we move forward in this sequence, the numbers approach closer to 0. Here we can say that the sequence converges to 0.

Limit as n approaches infinity of a n = 0

Let's look at a visual illustration:

N10.50

In the above figure, the red circles are getting closer to the x-axis as n increases, which shows convergence towards zero.

Another example of convergence

Let's consider another example that converges to a different limit:

2, 1.5, 1.333..., 1.25, 1.2, ...

Here, the sequence is governed by the rule a n = 2/n as n starts from 1. This sequence is converging to the value 1. Again, as n becomes larger, a n becomes closer to 1.

What is divergence?

If a sequence is not stationary at a single limit value it is said to diverge. This could mean that the numbers continue to get larger or smaller indefinitely, or they could oscillate without reaching a specific value. For example, consider this sequence:

1, 2, 3, 4, 5, ...

Here, the sequence continues indefinitely without reaching any finite limit. Thus, we say that this sequence diverges.

Divergent example with oscillation

Consider the sequence:

-1, 1, -1, 1, -1, 1, ...

The above sequence oscillates and does not reach any particular value. Hence, it also diverges.

Let's imagine a different sequence where the values keep increasing:

N0,

Notice how the blue circles represent numbers increasing without limit.

Understanding series convergence and divergence

When we consider whether a series converges or diverges, we look at the sum of the terms of the sequence. If the sum approaches a finite value as the number of terms increases, the series converges. If not, it diverges.

Example of convergent series

Consider the geometric series:

1 + 1/2 + 1/4 + 1/8 + 1/16 + ...

Each term is half of the previous term. As you add more terms, the sum approaches a limit. Specifically, this series converges to 2.

Mathematically it is expressed as:

S = 1 + 1/2 + 1/4 + 1/8 + ...
s = 2

Example of divergent series

Now consider the harmonic series:

1 + 1/2 + 1/3 + 1/4 + 1/5 + ...

Even though the terms get smaller, the sum of this series becomes infinitely large as more terms are added. Thus, the harmonic series diverges.

Visualization of series convergence and divergence

Let us understand the concept of convergent and divergent series:

Suppose we draw a graph where the x-axis represents the number of terms and the y-axis represents the value of the sum:

Number of postsApproaching a finite limit

The green line shows how a convergent series approaches a finite limit, while a divergent series continues to grow indefinitely, similar to the previously divergent sequence.

Criteria for convergence and divergence

There are several tests to determine whether a sequence or series is convergent or divergent:

  • Limit tests for sequences: If lim n→∞ a n = L exists and is finite, then the sequence converges on L. Otherwise, it diverges.
  • Testing the n-th term for divergence: If lim n→∞ a n ≠ 0, then the series Σa n diverges.
  • Geometric series test: A geometric series Σar n-1 converges if |r| < 1 and diverges otherwise.

Conclusion

Understanding the concepts of convergence and divergence is crucial for high-level mathematics and applications. These fundamental ideas help analysts predict behavior in mathematical systems that model the real world. Whether it is a sequence or a series, testing for convergence or divergence enables us to understand long-term consequences, making these concepts important tools in your mathematical journey.


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Convergence and Divergence

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