This Demonstration investigates the convergence (or otherwise) of the binomial series , which, when convergent, converges to the function . The output (in red) is shown in two ways:

(a) the partial sum of the series, for a chosen value of between and , as you vary the number of terms ;

(b) the graph (red) of the resulting polynomial function of , as you vary , in the interval .

Snapshot 1: is a nonnegative integer; the series terminates

Snapshot 2: is positive but not an integer; the series converges for

Snapshot 3: ; the series converges for

Snapshot 4: ; the series converges for

Snapshot 5: the graph of the truncated binomial series (a polynomial function of ) and the function for comparison, for the same value of and the same number of terms as in Snapshot 4

Reference

[1] T. Heard, D. Martin, and B. Murphy, A2 Further Pure Mathematics, 3rd ed., London: Hodder Education, 2005 pp. 69–83.