Well known series and sequences

[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

 

[NathanaelNolk]

I'd say the harmonic one ?

 

[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.

 

[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
$$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots$$

 

[NathanaelNolk]

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