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Sequences and Series
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Introduction
When beginning to explore sequences and series, it is helpful to look at the values of the terms in the sequence  and the values of the terms in the related sequence of partial sums  . Sometimes the graphs help to make the patterns more evident.
An example: 
We need to start by defining our function and naming it. Let  . Evaluate the cell below and each subsequent command as you go.
First, we will consider a numerical listing of the first 30 terms.
Next, view the same information graphically. There should not be any surprises here. The pattern is pretty clear.
After getting a sense of the terms of the sequence, we can add to this. We can also look at a table of the numerical values of  , the sequence of partial sums, along with n and  .
The next step will be to graph both  and  together. To do this, ask Mathematica to create each plot and then show them together.
The combination of  and  , either numerical or graphical, allows you to see more clearly how  is affected by  . Can you estimate the sum of the series? It turns out that the series converges to  . Look at this value below.
Seeing this, what can be said about the rate of convergence of the series? Clear the functions by executing the command below.
A different example: 
As before, we begin by giving the function a name. Call it h this time.
As in the first example, we will consider a numerical listing of the first 30 terms. What is the pattern?
We will follow the format of the example above and show the table of values with n,  , and  . How does the pattern for  vary from the first example?
Let us see the graphs.
Any guesses for the sum of the infinite series? It turns out to be  . Does this converge more quickly or more slowly than the series above? Clear the assigned names.
Next: 
We studied the corresponding improper integral carefully and we know that it diverged. Thus, it should be no surprise that the related series also diverges. First, define the function.
We will look at 30 terms again. You may want to edit the line to look at more terms.
Next, look at the sequence of partial sums.
The values of  increase very slowly. It is difficult to be very confident about the outcome of the series from these 30 values alone. Let us see the graphs.
Then clear the assigned names.
Your own example: 
This section will take you through the steps given in the first example above. Your first step is to create your own function. Complete the line below, expressing  in terms of k, and then evaluate the command. After that, evaluate any or all of the commands as your interest leads you. And of course, you can edit the lines to change the number of terms shown. For many functions, 30 terms seems to be fairly reasonable.
The command below shows a table of values of the sequence  .
Look at this same information graphically, rather than numerically.
Below, obtain a table of values with n,  , and  . Can you guess as to whether the series converges or diverges?
Here is the graph with your  and  together.
With the numerical values and the graphs, do you believe your series converges or diverges? How confident are you? Why? After viewing your graphs, clear the function and the plots. Then you can create another example and define it as f.
     
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