Calculus

The Particular Case of Positive Series

Consider the series and its associated sequence of partial sums . Here we will assume that the numbers we are about to add are positive, that is, for any...

The Geometric Series

Introduction Suppose someone offers you the following deal: You get $1 on the first day, $0.50 the second day, $0.25 the third day, and so on. For a second,...

Convergence of Series

Consider the series and its associated sequence of partial sums . We will say that is convergent if and only if the sequence is convergent. The total sum of...

Introduction to Series

The notion of series is closely related to the sum of numbers. In fact, whenever one hears the word series, the first thing to come to mind is the...

Problems on Sequences

In this page you will find some not so easy problems on sequences. Problem 1: Let be a sequence of real numbers such that . Show that . A...

Some Special Limits

Here we will discuss some important limits that everyone should be aware of. They are very useful in many branches of science. Example: Show using the Logarithmic function that...

Limit of a Sequence-2

Some basic properties. 1. The limit of a convergent sequence is unique. 2. Every convergent sequence is bounded. This is a quite interesting result since it implies that if...

Limit of a Sequence

The notion of limit of a sequence is very natural. Indeed, consider our scientist who is collecting data everyday. Set to be the sequence generated by our scientist (...

Sequences: Basic Definitions

Consider the sequence . It is clear that we have The numbers are getting bigger and bigger. Now consider the sequence . In this case, we have Notice that...

Introduction to Sequences

Sequences are to calculus what at calculator is to a scientist. There are many ways to introduce sequences. Here we will follow a somewhat unorthodox way. Indeed, consider a...