# Convergence of the Binomial Series

Convergence of the Binomial Series

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:

1+nx+n(n-1)/2!+…+n(n-1)…(n-r+1)/r!+…

2

x

r

x

n

(1+x)

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

x

-2

2

k

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

x

k

[-2,2]

The function is also shown for comparison (blue).

n

(1+x)