# Introduction to infinite series

##### Intros

###### Lessons

##### Examples

###### Free to Join!

StudyPug is a learning help platform covering math and science from grade 4 all the way to second year university. Our video tutorials, unlimited practice problems, and step-by-step explanations provide you or your child with all the help you need to master concepts. On top of that, it's fun — with achievements, customizable avatars, and awards to keep you motivated.

#### Easily See Your Progress

We track the progress you've made on a topic so you know what you've done. From the course view you can easily see what topics have what and the progress you've made on them. Fill the rings to completely master that section or mouse over the icon to see more details.#### Make Use of Our Learning Aids

#### Earn Achievements as You Learn

Make the most of your time as you use StudyPug to help you achieve your goals. Earn fun little badges the more you watch, practice, and use our service.#### Create and Customize Your Avatar

Play with our fun little avatar builder to create and customize your own avatar on StudyPug. Choose your face, eye colour, hair colour and style, and background. Unlock more options the more you use StudyPug.

###### 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 *

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

a) $\sum_{n=i}^{\infty}a_n+$ $\sum_{n=i}^{\infty}b_n=$$\sum_{n=i}^{\infty}(a_n+b_n)$

b) $\sum_{n=i}^{\infty}a_n-$ $\sum_{n=i}^{\infty}b_n=$$\sum_{n=i}^{\infty}(a_n-b_n)$

c) $\sum_{n=i}^{\infty}ca_n=$$c\sum_{n=i}^{\infty}a_n$

**Properties of Infinite Series:**If $\sum_{n=i}^{\infty}a_n$ and $\sum_{n=i}^{\infty}b_n$ are convergent series, then we can say that:

a) $\sum_{n=i}^{\infty}a_n+$ $\sum_{n=i}^{\infty}b_n=$$\sum_{n=i}^{\infty}(a_n+b_n)$

b) $\sum_{n=i}^{\infty}a_n-$ $\sum_{n=i}^{\infty}b_n=$$\sum_{n=i}^{\infty}(a_n-b_n)$

c) $\sum_{n=i}^{\infty}ca_n=$$c\sum_{n=i}^{\infty}a_n$

###### Basic Concepts

###### Related Concepts

2

videos

remaining today

remaining today

5

practice questions

remaining today

remaining today