Introduction to infinite series

Introduction to infinite series

In this lesson, we will talk about the concept of infinite series. We know that a normal series is the sum of all the terms of a finite sequence, but what about infinite series? Well, infinite series is the sum of all the terms of an infinite sequence. We will learn that not all infinite series add up to infinity. In fact, there are many infinite series which add up to a finite number. If we get a finite number, then we call the series convergent. Once we learn the concept, we will begin to talk about the properties of infinite series. These properties include adding and subtracting, and multiplying an infinite series by a constant. Lastly, we will talk about the index shift.
Basic Concepts: Introduction to sequences, Monotonic and bounded sequences
Related Concepts: Arithmetic series, Infinite geometric series

Lessons

Note *Properties of Infinite Series:

If ∑n=i∞an\sum_{n=i}^{\infty}a_n and ∑n=i∞bn\sum_{n=i}^{\infty}b_n are convergent series, then we can say that:

a) ∑n=i∞an+\sum_{n=i}^{\infty}a_n+ ∑n=i∞bn=\sum_{n=i}^{\infty}b_n=∑n=i∞(an+bn)\sum_{n=i}^{\infty}(a_n+b_n)
b) ∑n=i∞an−\sum_{n=i}^{\infty}a_n- ∑n=i∞bn=\sum_{n=i}^{\infty}b_n=∑n=i∞(an−bn)\sum_{n=i}^{\infty}(a_n-b_n)
c) ∑n=i∞can=\sum_{n=i}^{\infty}ca_n=c∑n=i∞anc\sum_{n=i}^{\infty}a_n
  • Introduction
    Overview of Infinite Series
    a)
    The Concept of Infinite Series

    b)
    Properties of Infinite Series

    c)
    Index Shift

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5.
Sequence and Series
5.1
Introduction to sequences
5.2
Monotonic and bounded sequences
5.3
Introduction to infinite series
5.4
Convergence and divergence of normal infinite series
5.5
Convergence & divergence of geometric series
5.6
Convergence & divergence of telescoping series
5.7
Divergence of harmonic series
5.8
P Series
5.9
Alternating series test
5.10
Divergence test
5.11
Comparison & limit comparison test
5.12
Integral test
5.13
Ratio test
5.14
Root test
5.15
Absolute & conditional convergence
5.16
Radius and interval of convergence with power series
5.17
Functions expressed as power series
5.18
Taylor series and Maclaurin series
5.19
Approximating functions with Taylor polynomials and error bounds
Calculus 2