Series with Interval of Convergence Dependent on Structure
This Demonstration shows an infinite series that, when written in two different ways, has a different interval of convergence.[more]
The power series expansion of is . The range of validity of the expansion, for which the series converges, is . However, if the series is written in the form , where , that is, , the interval of convergence is reduced to .
This Demonstration shows the difference between these two forms (the blue and purple graphs, respectively). It shows that beyond the value of , the second series diverges (the corresponding graph now changes color to red). The aim is to emphasize the fact that even though the first few terms and the sum to infinity (shown in green) of two infinite series may coincide, the actual structure of the series makes a difference with regard to the radius/interval of convergence.
(The notation <<>> indicates omitted terms.)[less]
 M. Spivak, Calculus, Cambridge: Cambridge University Press, 2006 pp. 464–490.