Write a power series as:

When , every term is except the first, so .

Differentiate the series:

Now set and what remains is .

Differentiate again and get , and evidently

So the series can be written in the form:

This is the **Maclaurin series**.

If the derivatives are evaluated at instead, we get the **Taylor series**:

**series for **

For the function etc.

At , since , the th derivative is just .

So the series can be represented:

When :

**series for **

When etc.

The derivatives at are repeating endlessly.

So the series is:

**series for **

When etc.

The derivatives at are repeating endlessly.

So the series is:

(or simply differentiating the series term by term gets the same series.)

**Euler’s identity**

Substitute into the series for to get:

Group the real and imaginary coefficients to get:

Plug in to get , or , “the most famous formula in all mathematics”.

A poll of readers conducted by The Mathematical Intelligencer in 1990 named Euler’s identity as the “most beautiful theorem in mathematics”. In another poll of readers…by Physics World in 2004, Euler’s identity tied with Maxwell’s equations (of electromagnetism) as the “greatest equation ever”. – wikipedia

Great article here explaining its significance intuitively.

**binomial expansion** of

Proceeding as above,

So the series can be written:

The coefficient of is the binomial coefficient , also written :

**Further reading**

Hamming, Richard – Methods of Mathematics, ch20-21