The following applet can be used to graph a sequence, a_n, and the corresponding series (i.e., the sequence of partial sums s_n = a₁ + a₂ + ... + a_n). Simply enter a_n, the function that gives the nth term of the sequence. Keep in mind that the applet plots a_n for n≥ 1, so be sure that a_n is defined for all positive integers n.

The user can also choose between Comparison Test, Limit Comparison Test, Ratio Test and Root Test to help determine the convergence of the original series.

Note that for large values of x-scale, the applet does not plot every element of the sequence. To force the applet to plot every element of the sequence, hold down the "SHIFT" key. Keep in mind that doing so may slow down the applet.

Here is a list of functions that can be used with this applet.

Use the above applet to determine:

• all the values of p ≥ 0 such that the sequence a_n = 1/n^p corresponds to a convergent series.
• all the values of r such that the sequence a_n = rⁿ corresponds to a convergent series.