Show that (1+ \frac{i}{n})^n \rightarrow e^i as n \rightarrow \infty for constant i.

Question:

Show that {eq}(1 +\frac{i}{n})^n \rightarrow e^i{/eq} as {eq}n \rightarrow \infty{/eq} for constant {eq}i{/eq}.

Existence of a Limit:

Suppose that {eq}a(x) {/eq} is a function that is defined in an interval that contains {eq}x=t {/eq}. Then limit is defined as:

{eq}\mathop {\lim }\limits_{x \to t} a(x) = M {/eq}

There exists a very small number {eq}k {/eq} such that {eq}k>0 {/eq} so that {eq}n>0 {/eq}. This means that {eq}|a(x) - M| < k {/eq} whenever {eq}0 < |x - t| < n {/eq}.

if {eq}\mathop {\lim }\limits_{x \to {t^ + }} a(x) = {A_1} {/eq} and {eq}\mathop {\lim }\limits_{x \to {t^ - }} a(x) = {A_2} {/eq}, then;

(1) Limit exist if,

{eq}{A_1} = {A_2} {/eq}

(2) Limit does not exist if,

{eq}{A_1} \ne {A_2} {/eq}

Answer and Explanation:

Given that: {eq}\displaystyle \mathop {\lim }\limits_{n \to \infty } {\left( {1 + \frac{i}{n}} \right)^n} {/eq}

{eq}\displaystyle \eqalign{ & \mathop {\lim }\limits_{n \to \infty } {\left( {1 + \frac{i}{n}} \right)^n} \cr & \mathop {\lim }\limits_{n \to \infty } {\left( {1 + \frac{i}{n}} \right)^n}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{1^\infty }{\text{ form}}} \right) \cr & {e^{\mathop {\lim }\limits_{n \to \infty } n\left( {1 + \frac{i}{n} - 1} \right)}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\mathop {\lim }\limits_{x \to a} {{(f(x))}^{g(x)}} = {e^{\mathop {\lim }\limits_{x \to a} g(x)(f(x) - 1)}},(f(a) = 1,g(a) = \infty )} \right) \cr & {e^{\mathop {\lim }\limits_{n \to \infty } n\left( {\frac{i}{n}} \right)}} \cr & {e^{\mathop {\lim }\limits_{n \to \infty } i}} \cr & {e^i} \cr & \cr & \mathop {\lim }\limits_{n \to \infty } {\left( {1 + \frac{i}{n}} \right)^n} = {e^i} \cr} {/eq}


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Understanding the Properties of Limits

from Math 104: Calculus

Chapter 6 / Lesson 5
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