1. Power series
is called a power series with the centre at x0 or centred around x0
A power series centred around zero is a special case:

2. When does a power series
define a function?
In other words, for which x the resulting number series
converges ?
It certainly does at x = x0 since

3. If a power series converges at a point , then it converges absolutely in the open interval and converges uniformly in each closed interval
where a x0

4. Note: if a power series only converges at its center, we put r = 0.
Unless a power series converges at any real number, a number r > 0 exists such that converges absolutely for each x such that and diverges for any other x
This number is called the radius of convergence and
is the interval of convergence of
Note: if a power series only converges at its center, we put r = 0.

5. Let be a power series and let
Then for the radius of convergence r of we have
r =
Note that if an has a limit, then

6. Example
What is the radius of convergence of the power series ?

7. Example
What is the radius of convergence of the power series ?
Let k be an arbitrary natural number. Then, for n > k we have
. This means that
Since , we have for any k, which means that and
Thus

8. Derivative of a power series
If we differentiate each term of the power series , we obtain a new power series
This new power series is then called the derivative of a given power series.

9. The power series
and its derivative
have the same radius of convergence

10. If then
for where r is the radius of convergence of both the power series and

11. Since is again a power series with the identical radius of convergence r it has again a derivative in the convergence interval (r, r). This derivative is again a power series with the same radius of convergence r and so on:
has derivatives of arbitrary order in its
convergence interval (r, r)

12. Example
We know that for
Differentiating the series term by term yields:

13. Integral of a power series
Let be a power series with r as the radius of convergence. We can integrate it term by term from 1 to x where obtaining
The radius of convergence of the resulting series is again r.

14. Example
If we integrate term by term the power series
from 0 to x where , we obtain the power series
Note that the resulting power series converges even for x = 1 but not for x = 1.

15. Example
If we integrate term by term the power series
from 0 to x where , we obtain the power series
Note that the resulting power series converges even for x = 1 and x = 1.

16. Calculating
The preceding formula can be used for calculating numerically the constant
However, this series converges very slowly and is therefore not at all suitable for numerical calculation. To reach a reasonable accuracy of several decimal places probably several billions terms would have to be calculated.

17. The reason is that we have to use x at the end of the convergence interval. For practical calculations, better schemes have been devised such as:
This identity was discovered by Machin in the 18th century.
pocket calculator value

18. How Machin's identity can be derived:x
This can be verified by calculation