# Infinite Series

The **sum** of infinite terms that follow a rule.

When we have an infinite sequence of values:

\frac{1}{2} , \frac{1}{4} , \frac{1}{8} , \frac{1}{16} , ...

which follow a rule (in this case each term is half the previous one),

and we **add them all up**:

\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + ... = S

we get an **infinite series**.

"Series" sounds like it is the **list of numbers**, but it is actually when we **add them up**.

*(Note: The dots "..." mean "continuing on indefinitely") *

## First Example

You might think it is impossible to work out the answer, but sometimes it can be done!

Using the example from above:

\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + ... = 1

And here is why:

(We also show a proof using Algebra below)

## Notation

We often use Sigma Notation for infinite series. Our example from above looks like:

This symbol (called Sigma) means "sum up" |

Try putting 1/2^n into the Sigma Calculator.

## Another Example

\frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \frac{1}{256} + ... = \frac{1}{3}

Each term is a quarter of the previous one, and the sum equals 1/3:

Of the 3 spaces (1, 2 and 3) only number 2 gets filled up, hence 1/3.

## Converge

Let's add the terms one at a time. When the "sum so far" approaches a finite value, the series is said to be "**convergent**":

Our first example:

\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + ...

Adds up like this:

Term | Sum so far | |

1/2 | 0.5 | |

1/4 | 0.75 | |

1/8 | 0.875 | |

1/16 | 0.9375 | |

1/32 | 0.96875 | |

... | ... |

The sums are heading towards 1, so this series is **convergent**.

The "sum so far" is called a partial sum .

So, more formally, we say it is a convergent series when:

"the sequence of partial sums has a finite limit."

## Diverge

If the sums do not converge, the series is said to **diverge**.

It can go to **+infinity**, **−infinity** or just go up and down without settling on any value.

### Example:
1 + 2 + 3 + 4 + ...

Adds up like this:

Term | Sum so far | |

1 | 1 | |

2 | 3 | |

3 | 6 | |

4 | 10 | |

5 | 15 | |

... | ... |

The sums are just getting larger and larger, not heading to any finite value.

It does not converge, so it is **divergent**, and heads to infinity.

### Example: 1 − 1 + 1 − 1 + 1 ...

It goes up and down without settling towards some value, so it is **divergent**.

## More Examples

### Arithmetic Series

When the **difference** between each term and the next is a constant, it is called an **arithmetic series**.

(The difference between each term is 2.)

### Geometric Series

When the **ratio** between each term and the next is a constant, it is called a **geometric series**.

Our first example from above is a geometric series:

(The ratio between each term is **½**)

And, as promised, we can show you why that series equals 1 using Algebra:

**First**, we will call the whole sum

**"S"**: S = 1/2 + 1/4 + 1/8 + 1/16 + ...

**Next**, divide

**S**by

**2**:S/2 = 1/4 + 1/8 + 1/16 + 1/32 + ...

Now **subtract** S/2 from S

All the terms from 1/4 onwards cancel out.

### Harmonic Series

This is the Harmonic Series:

It is divergent.

How do we know? Let's compare it to another series:

1 | + | 1 | + | 1 | + | 1 | + | 1 | + | 1 | + | 1 | + | 1 | + | 1 | + ... | ||

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||||||||||||

etc... | |||||||||||||||||||

1 | + | 1 | + | 1 | + | 1 | + | 1 | + | 1 | + | 1 | + | 1 | + | 1 | + ... | ||

2 | 4 | 4 | 8 | 8 | 8 | 8 | 16 |

In each case, the **top values are equal or greater** than the bottom ones.

Now, let's add up the bottom groups:

1 | + | 1 | + | 1 | + | 1 | + | 1 | + | 1 | + | 1 | + | 1 | + | 1 | ... | + ... | ||

2 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | |||||||||||||

1 | + | 1 | + | 1 | + | 1 | + | 1 | + ... = ∞ | |||||||||||

2 | 2 | 2 | 2 |

That series is divergent.

So the harmonic series must also be divergent.

Here is another way:

We can sketch the area of each term and compare it to the area under the **1/x** curve:

**1/x** vs harmonic series area

Calculus tells us the area under 1/x (from 1 onwards) approaches **infinity**, and the harmonic series is greater than that, so it must be divergent.

### Alternating Series

An example of an Alternating Series (based on the Harmonic Series above):

It moves up and down, but in this case converges on the natural logarithm of 2

### Advanced Explanation:

To show WHY, first we start with a square of area 1, and then pair up the minus and plus fractions to show how they cut the area down to the area under the curve **y=1/x** between 1 and 2:

Can you see what remains is the area of 1/x from 1 to 2?

Using integral calculus (trust me) that area is **ln(2)**:

(As an interesting exercise, see if those rectangles really do make the area shown!)

## More

There are other types of Infinite Series, and it is interesting (and often challenging!) to work out if they are convergent or not, and what they may converge to.