To explain infinite series using sigma notation, we first need to know what a series is.

An infinite series happens when we keep adding the terms of a sequence forever.

In math, we show a sequence with ( a_n ). The letter ( n ) usually stands for a counting number, starting from 1, 2, 3, and so on.

The special symbol we use for adding is ( \Sigma ). If we want to write a series that starts from ( n = 1 ), we can show it like this:

$$\sum_{n=1}^{\infty} a_n$$

This expression simply means that we are adding the terms ( a_1, a_2, a_3, \ldots ) forever.

Parts of Sigma Notation:

1. Summation Symbol: ( \Sigma ) shows that we are adding things together.
2. Index of Summation: ( n ) is the number that changes as we add more terms. It starts at a certain number and goes up.
3. Limit of Summation: The upper limit here is ( \infty ), which means the series goes on without stopping.
4. General Term: ( a_n ) is the formula that tells us what the ( n )-th term is.

Examples:

• One common series is the geometric series:

$$\sum_{n=0}^{\infty} ar^n$$

In this, ( a ) is the first term, and ( r ) is how much we multiply each term by.

• Another example is the harmonic series:

$$\sum_{n=1}^{\infty} \frac{1}{n}$$

Learning about sigma notation helps us work with infinite series easily. It lets us analyze them, check if they converge (or come to a limit), and solve problems in calculus.