Convergence of a sequence

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

Enter a rule for the sequence $f_n$ in the box provided.

Drag the green point to adjust the value of L.

Click Show ε or Show M to display points for ε and M, and their corresponding regions. For each ε, can you find an M so that all points in the blue region are also in the orange region?

Zoom in or out using the buttons, if needed.

You can add a second sequence for comparison by enabling 'Show second sequence'.

Some interesting sequences to try:

  • $f_n = (-1)^n$ – enter this as $f_n$ = (-1)^n
  • $f_n = (-1)^n/n$ – enter this as $f_n$ = (-1)^n/n
  • $f_n = \sin(n)$
  • $f_n = (1+1/n)^n$ – enter this as $f_n$ = (1+1/n)^n

Other resources:

Instructors' notes for this applet

Geogebratube page for this applet