# Well known series and sequences

1. Jul 17, 2014

### fog37

Hello Forum,

I am familiar with the arithmetic sequence (the difference between one entry and the previous one is constant) and the geometric sequence ( the ratio between one entry and the previous one is constant).

are there any other important and simple sequences I should be aware of?

There is also the arithmetic and geometric series. Each one is the summation of terms from the arithmetic and geometric sequences respectively, correct?

thanks
fog37

2. Jul 17, 2014

### NathanaelNolk

I'd say the harmonic one ?

3. Jul 17, 2014

### fog37

Sure!

The harmonic is another one: 1, 1/2, 1/3, etc...
The reciprocal of terms of a harmonic sequence form an arithmetic sequence. I guess this the principle that allows us to determine if a sequence is harmonic or not, i.e. we take the reciprocals and test if their difference is a constant along the whole sequence....

What about the hypergeometric sequence? Does it exist? I have heard of the hypergeometric series which I presume to be the summation of the terms of a hypergeometric sequence.

4. Jul 17, 2014

### jbunniii

Power series in general are quite important and provide many concrete examples. An important one is the exponential function:
$$\exp(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$
Evaluating this at $x=1$, we get a series converging to $e$:
$$e = \sum_{n=0}^{\infty} \frac{1}{n!}$$
And here's a sequence which also converges to $e$:
$$e = \lim_{n\rightarrow \infty} \left(1 + \frac{1}{n}\right)^n$$
The arctangent can also expressed as a power series:
$$\arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$$
Evaluating at $x=1$, we get
$$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots$$

5. Jul 17, 2014

### NathanaelNolk

Correct me if I'm wrong but the hypergeometric series is also a power series.