Infinite Series

The sum of infinite terms that follow a rule.

When we have an infinite sequence of values:

12 , 14 , 18 , 116 , ...

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

and we add them all up:

12 + 14 + 18 + 116 + ... = 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:

12 + 14 + 18 + 116 + ... = 1

And here is why:

Sum of 1/2^n as boxes
(We also show a proof using Algebra below)

Notation

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

Sum of 1/2^n

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

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

Another Example

14 + 116 + 164 + 1256 + ... = 13

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

Sum 1/4^n as boxes
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:

12 + 14 + 18 + 116 + ...

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.

In calculus we say "the sequence of partial sums has a finite limit."

Diverge

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

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 the series is divergent.

Example: 1 − 1 + 1 − 1 + 1 ...

It goes up and down without settling towards some value, so it diverges.

More Examples

Arithmetic Series

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

Sigma n=0 to infinity of (10+2n) = 10+12+14+...

(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:

Sum of 1/2^n

(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 them!

All the terms from 1/4 onwards cancel out.

And we get:S − S/2 = 1/2
Simplify: S/2 = 1/2
And so:S = 1

 

Harmonic Series

This is the Harmonic Series:

Sigma n=1 to infinity of (1/n) = 1 + 1/2 + 1/3 + 1/4 + ...

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
down   down   down   down   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
down   down   down   down   down  
1 +   1     +     1     +         1         +     1   + ... = ∞
  2     2         2         2

So our original series must also be infinite.

 

Alternating Series

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

Sigma n=1 to infinity of (-1)^(n+1) /n = 1 - 1/2 + 1/3 - 1/4 + ... = ln(2)

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:

alternating harmonic proof

Using integral calculus (trust me) the area under that y=1/x curve is ln(2):

   2
1
1/x dx = ln(2) − ln(1) = ln(2)

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

More

There are more 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.