Proof that exp(z) = lim (1 + z/n)^n as n goes to infinity

For all complex numbers, z, e^z is equal to the limit of (1 + z/n)^n, as n approaches infinity.

Category:

Education

Tags:

• analysis
• base
• of
• natural
• logarithms
• exponent
• complex
• variables
• mathematics
• proof

[cwldoc]

There are various ways of defining the exponential function on the complex plane. One definition used is

exp(z) = 1 + z + (z^2)/2! + . . . + (z^n)/n! + . . .

and another equivalent definition is the limit mentioned in the title. Yet another equivalent definition is hinted at in the last post:

Define exp(z) to be the function whose derivative is itself and whose value at 0 is 1.

[anticorncob6]

e = 1 + 1 + 1/2! + 1/3! + 1/4!...

e^x = x + x^2/2! + x^3/3!...

e = limit as n approaches infinity (1+1/n)^n

e^x = limit as x approaches infinity (1+x/n)^n as n approaches infinity

When you're dealing with e, there's two ways to write things: as limits and as infinite series. When you prove that the derivative of e^x is e^x, those two ways are available (although the series expansion is much easier to prove than the limits)