Infinite Series

Definition of Infinite Series

Infinite series are the sum of infinitely many numbers listed in a given order & related in a given way.

For an infinite series a1 + a2 + a3 + … , a quantity sn = a1 + a2 + … + an, which involves adding only the first n terms, is called a partial sum.

For Sn approaches a fixed number S as n becomes larger, the series is said to converge. Where, S is called the sum of the series.

If the infinite series is not converge, it is said to diverge. Where there’s no value of a sum is assigned.

Infinite Series: The sum of infinite terms that follow a rule.

When we have an infinite sequence of values:

Infinite Series Example

which follow a rule (in this case each term is half the previous one), and we add them all up:

we get an infinite series.

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

*Note: The dots “…” mean “continuing on indefinitely”


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

Using the example from above:

Infinite Series Example

And here is why:

Infinite Series Illustration

There are a lot of mathematical problems that use complicated function that can be solved easily and directly if the function can be expressed as an infinite series that involving trigonometric functions (such as sine & cosine).

The process of breaking up arbitrary function into an infinite trigonometric series is called harmonic analysis or Fourier analysis. It has many applications in the study of various wave phenomena.


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

Infinite Series Notation

Σ (called Sigma) means “sum up”


When the “sum so far” approaches a finite value, the series is said to be “convergent”.

Infinite Series Example

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

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

“the sequence of partial sums has a finite limit.”


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.

2 + 4 + 6 + 8 + 16 + …

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.

Arithmetic Series

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

2 + 4 + 6 + 8 + 16 + …

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.

Infinite Series Example

The ratio between each term is ½.

Harmonic Series

This is the Harmonic Series:

Infinite Harmonic Series

It is divergent.

Alternating Series

An Alternating Series has terms that alternate between positive and negative.

Infinite Alternating Series

Alternating Series can be converge or not.


Sometimes we can get weird results when we change the order.

For example in an alternating series, what if we made all positive terms come first? So The order of the terms can be very important!

Learn More

Arithmetic Sequences and Series Sums

Geometric Sequences and Sums

Trigonometric Identities

Parallel and Perpendicular Lines

Algebra Index