What is the difference between arithmetic and geometrical series? What is the difference between arithmetic and geometrical series?
Also what are they?
How do they look like?
 A: Geometric and arithmetic are two names that are given to different sequences that follow a rather strict pattern for how one term follows from the one before.
An arithmetic sequence is characterised by the fact that every term is equal to the term before plus some fixed constant, called the difference of the sequence. For instance,
$$
1,4,7,10,13,\ldots
$$
is an arithmetic sequence with difference $3$, while
$$
1,2,4,6,7,10,\ldots
$$
is not an arithmetic sequence.
A geometric sequence follows a very similar idea, except instead of adding a fixed number to get from one term to the next, you multiply by a number, called the quotient of the sequence. For instance,
$$
1,2,4,8,16,32,\ldots
$$
is a geometric sequence with quotient $2$, while
$$
3,6,13,23,48,\ldots
$$
fails to be a geometric sequence.
The word "series" is usually used to signify a sum of consecutive terms in a sequence. In the case of arithmetic series the sum is almost always of a finite number of terms. (A commonly mentioned exception is $1+2+3+4+\cdots =-\frac{1}{12}$, which while apparently just nonsense, has been verified experimentally in quantum mechanics. Specifically in the context of the Casimir effect.)
In a geometric series, if the quotient is between $-1$ and $1$, one can take the sum of all the infinitely many terms of the sequence. A notable example one should be familiar with is
$$
\frac12 + \frac14+\frac18+\cdots =1
$$
