I've been wondering if/when it's possible to "truncate" a series.
Example 1
For example, the closed form for the series of naturals is:
$\frac{1}{(x-1)^2}$ = $1 + 2z + 3x^2 + \cdots$
The "truncated" form is:
$\frac{1-(n+1)x^n+n\cdot x^{n+1}}{(x-1)^2}$ = $1 + 2z + 3x^2 + \cdots + n\cdot x^{n-1}$
Example 2
The closed form of integers of the form $2^n$ is:
$\frac{1}{1-2x}$ = $1 + 2x + 4x^2 + \cdots + (2x)^n + \cdots$
The "truncated" form is:
$\frac{1 - (2x)^n}{1 - 2x}$ = $1 + 2x + 4x^2 + \cdots + (2x)^{n-1}$
The Search
Could anyone help with finding appropriate resources that address "truncating" a series? It would be great if there is some work that may include math that relates to finding truncated versions of series.
I don't have a very good background in math, so anything that doesn't require a lot of background would be very beneficial. However, I am very interested in learning, and ANY resource that touches upon this topic would be very beneficial and interesting for me.
I'd just like to know where I can learn more about this and related topics. Again, I'm very interested in being able to produce a closed form or recurrence for a "piece" of a series.