Polynomial Approximation of the Exponential Function

n

2

The graph shows plots of

y=exp(x)

(dashed line) and

y=

n

(1+x/n)

for various values of

n

. It is interesting that for positive values of

n

, the latter expression is a polynomial that converges from below to

exp(x)

(the blue and violet lines are the polynomials). For negative values of

n

, the expression is the reciprocal of a polynomial that converges to

exp(x)

from above (the green, yellow, and orange curves are the reciprocals of polynomials).