Convergence of the Binomial Series

output

sum

graph

exponent n

0.5

number of terms k

20

x

0.5

This Demonstration investigates the convergence (or otherwise) of the binomial series

1+nx+n(n-1)/2!

2

x

+…+n(n-1)…(n-r+1)/r!

r

x

+…

, which, when convergent, converges to the function

n

(1+x)

. The output (in red) is shown in two ways:

(a) the partial sum of the series, for a chosen value of

x

between

-2

and

2

, as you vary the number of terms

k

;

(b) the graph (red) of the resulting polynomial function of

x

, as you vary

k

, in the interval

[-2,2]

.

The function

n

(1+x)

is also shown for comparison (blue).