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# Intuitive Guide on Euler's Number

*schedule*Aug 12, 2023

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This guide requires that you are familiar with the concept of compound interests. If not, please consult our guide on compound interests.

# Motivating example

Recall that the formula to compute compound interest:

Where:

$A$ is the final amount in the bank account after $t$ years.

$P$ is the principal, that is, the starting amount in the bank account.

$r$ is the interest rate (e.g. $0.1$).

$n$ is the number of compounds per year.

$t$ is the number of years elapsed.

Now, consider a simple case with the following assumptions:

we initially start with one dollar in the bank account, that is, $P=1$.

the interest rate is $100\%$, that is, the bank will double our money at every compound period. This would mean $r=1$.

we are interested in our bank balance after one year, that is, $t=1$.

For this simple case, our formula for compound interest becomes:

The only variable here is $n$, which is the number of compounds per year. For instance, $n=1$ for annual compounds, whereas $n=4$ for quarterly compounds. Let's now compute $A$ for different values of $n$ like so:

$n$ | $A$ | $n$ | $A$ |
---|---|---|---|

1 (annually) | 2.0000 | 52 (weekly) | 2.6926 |

2 (bi-annually) | 2.2500 | 365 (daily) | 2.7146 |

4 (quarterly) | 2.4414 | 8760 (hourly) | 2.7181 |

12 (monthly) | 2.6130 | 525600 (per min) | 2.7183 |

Note that the values of $A$ are all rounded to four decimal places. We can see that as $n$ increases, the value of $A$ also increases, but the amount of increase slows down. For instance, increasing $n$ from $1$ to $2$ gives us an additional $\$0.25$, but increasing $n$ from $365$ to $8760$ only gives us around $\$0.003$ additionally.

As a result, $A$ seems to be converging to some constant of around $2.718$; this constant is the famous Euler's constant $e$. The larger the $n$, the closer we approach $e$, hence it makes sense to define $e$ as the limit of $A$ as $n$ tends to infinity.

# Formal definition of Euler's constant

Euler's constant $e$ is defined as follows:

Note that $e$ is irrational:

Let's now expand on the formal definition of Euler's constant and express $e^\lambda$ as a limit where $\lambda$ is some scalar number.

# Expressing the power of Euler's constant as limits

If $\lambda\in\mathbb{R}$, then:

Proof. From the basic rules of taking powers, we have that:

Now, let's take the limit as $n$ tends to infinity:

We now want to show that the right-hand side is equal to $e^\lambda$. Let's define a new variable:

Notice how $k$ also tends to infinity as $n$ tends to infinity. Rewriting the right-hand side of \eqref{eq:yTGF12ryu589R5LxGty} in terms of $k$ gives:

Using the power rule of limits, we can take the power of $\lambda$ outside the limit:

Substituting \eqref{eq:PxSqujAPFRRw0WW1Wlc} into \eqref{eq:yTGF12ryu589R5LxGty} gives us the desired identity:

This completes the proof.