Introduction to infinite series

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  1. Overview of Infinite Series
  2. The Concept of Infinite Series
  3. Properties of Infinite Series
  4. Index Shift
Topic Notes
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.
Note *Properties of Infinite Series:

If n=ian\sum_{n=i}^{\infty}a_n and n=ibn\sum_{n=i}^{\infty}b_n are convergent series, then we can say that:

a) n=ian+\sum_{n=i}^{\infty}a_n+ n=ibn=\sum_{n=i}^{\infty}b_n=n=i(an+bn)\sum_{n=i}^{\infty}(a_n+b_n)
b) n=ian\sum_{n=i}^{\infty}a_n- n=ibn=\sum_{n=i}^{\infty}b_n=n=i(anbn)\sum_{n=i}^{\infty}(a_n-b_n)
c) n=ican=\sum_{n=i}^{\infty}ca_n=cn=ianc\sum_{n=i}^{\infty}a_n