Note: I've deleted all of the output in the following in order to keep the file size small and so you can see the output created for yourself.  To do this, put the cursor in each input cell and hit Enter or Shift-Return.

We'll use the Sum command to build our partial sums.  Here's an example.:

Sum[x^n,{n,0,10}]

Here's a plot of this specific partial sum:

Plot[Sum[x^n,{n,0,10}],{x,-1.5,1.5},PlotRange->{-10,10}]

In the Plot command above, I used the PlotRange option to force a specific vertical range.  I want to do this below so that the frames of the movie we'll make all have the same window.

The next bit of input uses the Table command to make many plots.  Notice that the iterator nmax for the Table command is used as the upper value for the Sum iterator.  Also, notice that I have named the output myframes so I can refer to this in later work.  Also notice that I've entered this input with linebreaks to help you read the syntax.

myframes=
Table[
Plot[
Sum[x^n,{n,0,nmax}],
{x,-1.5,1.5},
PlotRange->{-10,10}],
{nmax,0,20}]

With a list of pictures, we can use the ListAnimate command to make a movie:

ListAnimate[myframes]

We'll now make a new movie that includes the limit function in each frame.  First, make and name a plot of the limit function:

limitplot=
Plot[1/(1-x),{x,-1,1},PlotStyle->RGBColor[1,0,0],PlotRange->{0,10}]

The Show command can be used to combine pictures.  Here's an example:

Show[myframes[[1]],limitplot]

In the example above, the syntax myframes[[1]] refers to the first element of the list that we've called myframes.

We'll use Show inside a Table command to make a new list of frames, each of which includes limitplot.  I'm also including a piece of text in each frame.  This is created by the code 
Graphics[Text["N=" StandardForm[n], {1, -5}]]
So, each Show is combining three things: an element of myframes, the plot limitplot, and a piece of text.

mynewframes=
Table[
Show[myframes[[n+1]],
limitplot,
Graphics[Text["N="StandardForm[n],{1,-5}]
]
],
{n,0,Length[myframes]-1}]

Finally, another use of ListAnimate gets us the new movie:

ListAnimate[mynewframes]