This applet illustrates the ε-M definition of convergence of a sequence.

Enter a rule for the sequence [math]f_n[/math] in the box provided.[br][br]Drag the [color=#006400]green[/color] point to adjust the value of [b][color=#006400]L[/color][/b].[br][br]Click '[b][color=#ff7f00]Show ε[/color][/b]' or '[b][color=#0000ff]Show M[/color][/b]' to display points for [b][color=#ff7f00]ε[/color][/b] and [b][color=#0000ff]M[/color][/b], and their corresponding regions. For each [b][color=#ff7f00]ε[/color][/b], can you find an [b][color=#0000ff]M[/color][/b] so that all points in the [color=#0000ff]blue[/color] region are also in the [color=#ff7f00]orange [/color]region?[br][br]Zoom in or out using the buttons, if needed.[br][br]You can add a second sequence for comparison by enabling 'Show second sequence'.[br][br][br]Some interesting sequences to try:[br][list][br][*] [math]f_n = (-1)^n[/math] - enter this as [math]f_n[/math] = (-1)^n[br][*] [math]f_n = (-1)^n/n[/math] - enter this as [math]f_n[/math] = (-1)^n/n[br][*] [math]f_n = \sin(n)[/math][br][*] [math]f_n = (1+1/n)^n[/math] - enter this as [math]f_n[/math] = (1+1/n)^n[br][/list]