http://www.physics.miami.edu/~nearing/mathmethods/series.pdf

## Series worth knowing

**Geometric Series**

Consider:

so

**Power Series**

- for
- for (Binomial series)

**Taylor Series**

## Convergence Tests

**Comparison test**

Let be sequences of positive reals and s.t. and converges, then converges

**Ratio test**

If for large , , then converges. Derived from comparison test with geometric series.

**Integral test**

If is *decreasing positive* and converges, converges

## Stirling’s approximation

Using newton’s method to approximate this to the 2nd order:

(for moderately large *n*)

The last step in the above makes use of the **Gaussian integral**:

using **parametric differentiation**: