The Binomial Expansion

The red function is to be approximated using the Binomial expansion. You can change this function using the green and purple sliders. The blue function is the polynomial that comes from the Binomial expansion. Increase the number of terms it contains (its degree) using the blue slider. Where is the approximation most accurate? The expansion is completely invalid outside of the "radius of convergence". Tick the box to see this. When does the radius of convergence change? Can you find a function that can be matched exactly by the Binomial expansion, for all values of x?