The Number e & Binomial Expansion
What happens if a bank pays you interest on your interest on your interest, over and over, faster and faster? You might expect your money to grow forever, but it actually settles down to a specific magical number: \(e \approx 2.71828\). In this lesson, you’ll discover where \(e\) comes from, learn the binomial expansion trick that unlocks it, and see why \(e\) is one of the most important numbers in all of math.
Imagine you put $1 in a bank account that pays 100% interest per year. If the bank compounds once a year, you get $2. But what if they compound every month? Every day? Every second? As you compound more and more frequently, the amount you end up with gets closer and closer to a mysterious number: \(e \approx 2.71828...\). This number shows up everywhere — in population growth, radioactive decay, and even in how social media posts go viral. Today we will see exactly where it comes from and why it is so special.
Topics Covered
- The origin of the number \(e\) through compound interest
- The limit \(\left(1 + \frac{r}{k}\right)^k \to e^r\) as \(k \to \infty\)
- Why the derivative of \(e^x\) is \(e^x\) itself — the defining property
- Binomial expansion: \((a+b)^n = \sum_{i=0}^{n} \binom{n}{i} \cdot a^{n-i} \cdot b^{i}\)
- Combinatorics review: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)
- Complementarity: \(\binom{n}{k} = \binom{n}{n-k}\)
- Patterns in binomial coefficients that lead to the limit definition of \(e\)
Lecture Video
Key Frames from the Lecture
You should be comfortable with the basic laws of exponents:
- \(a^m \cdot a^n = a^{m+n}\)
- \((a^m)^n = a^{mn}\)
- \(a^0 = 1\) for any \(a \neq 0\)
These rules are essential for understanding how \((1 + r/k)^k\) behaves as we change \(k\).
A factorial is written \(n!\) and means the product of all positive integers up to \(n\):
\[n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1\]
For example: \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
By convention, \(0! = 1\). Factorials grow incredibly fast — \(10! = 3{,}628{,}800\) is already over three million.
The Origin of the Number \(e\): Compound Interest
Suppose you invest $1 at 100% annual interest rate (\(r = 1\)). If the bank compounds \(k\) times per year, each compounding period gives you a rate of \(\frac{1}{k}\), and after one year you have:
\[A = \left(1 + \frac{1}{k}\right)^k\]
Let us see what happens as \(k\) grows:
| Compounding (\(k\)) | Expression | Value |
|---|---|---|
| 1 (annually) | \((1 + 1)^1\) | \(2.000\) |
| 2 (semi-annually) | \((1 + 0.5)^2\) | \(2.250\) |
| 12 (monthly) | \((1 + 1/12)^{12}\) | \(2.613...\) |
| 365 (daily) | \((1 + 1/365)^{365}\) | \(2.7146...\) |
| 10,000 | \((1 + 1/10000)^{10000}\) | \(2.71815...\) |
| \(\infty\) | \(\lim_{k \to \infty} (1 + 1/k)^k\) | \(e \approx 2.71828...\) |
The number never blows up to infinity — it settles down to the special constant \(e\).
Explore the limit — use the slider to increase \(k\) and watch the value approach \(e\):
Start with \(k = 1\) and slowly increase it. Notice how quickly the value gets close to \(e\) — by \(k = 100\) you are already within a few hundredths. But it never quite reaches \(e\); it just gets closer and closer forever. That is what a limit means.
Generalizing: The Rate \(r\)
What if the interest rate is not 100% but some general rate \(r\)? Then after one year with \(k\) compoundings:
\[A = \left(1 + \frac{r}{k}\right)^k\]
As \(k \to \infty\), this approaches \(e^r\):
\[\lim_{k \to \infty} \left(1 + \frac{r}{k}\right)^k = e^r\]
This is one of the most important limits in calculus. But to prove it, we need a powerful algebraic tool: the binomial expansion.
Combinatorics Review: \(n\) Choose \(k\)
Before we can expand \((a + b)^n\), we need the binomial coefficient, read “\(n\) choose \(k\)”:
\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
This counts the number of ways to choose \(k\) items from a group of \(n\) items, where order does not matter.
Example: How many ways can you pick 3 students from a class of 5?
\[\binom{5}{3} = \frac{5!}{3! \cdot 2!} = \frac{120}{6 \cdot 2} = 10\]
Complementarity: \(\binom{n}{k} = \binom{n}{n-k}\)
Here is a beautiful fact: choosing which items to take is the same as choosing which items to leave behind.
\[\binom{100}{96} = \binom{100}{4}\]
Think about it: picking 96 people from 100 to be on a team is the same as picking 4 people to not be on the team. Instead of listing 96 names, you just list the 4 who sit out. Same number of ways, much less writing!
This is why \(\binom{n}{k} = \binom{n}{n-k}\) — the formula is symmetric.
\[\binom{n}{n-k} = \frac{n!}{(n-k)!\left(n-(n-k)\right)!} = \frac{n!}{(n-k)! \cdot k!}\]
That is exactly the same as \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), just with the two factors in the denominator swapped. Multiplication is commutative, so they are equal.
The Binomial Expansion
Now for the main tool. The Binomial Theorem says:
The binomial theorem lets you expand any power of a sum \((a+b)^n\) into individual terms. It is the algebraic engine that powers our proof of the compound interest limit.
\[(a + b)^n = \sum_{i=0}^{n} \binom{n}{i} \cdot a^{n-i} \cdot b^{i}\]
This means we expand \((a+b)^n\) by summing over all ways to pick some copies of \(b\) and the rest from \(a\).
Example: Expand \((a + b)^3\)
\[(a+b)^3 = \binom{3}{0}a^3 + \binom{3}{1}a^2 b + \binom{3}{2}a b^2 + \binom{3}{3}b^3\]
\[= a^3 + 3a^2b + 3ab^2 + b^3\]
Example: Expand \((x + 1)^4\)
\[(x+1)^4 = \binom{4}{0}x^4 + \binom{4}{1}x^3 + \binom{4}{2}x^2 + \binom{4}{3}x + \binom{4}{4}\]
\[= x^4 + 4x^3 + 6x^2 + 4x + 1\]
Notice the coefficients \(1, 4, 6, 4, 1\) — these are the 4th row of Pascal’s Triangle.
Patterns in the Binomial Coefficients
When we write out \(\binom{n}{k}\) for a general \(n\), something useful happens. Let us write the first few coefficients with \(k\) factors on top and \(k!\) on the bottom:
\[\binom{n}{0} = 1, \qquad \binom{n}{1} = \frac{n}{1}, \qquad \binom{n}{2} = \frac{n(n-1)}{2!}, \qquad \binom{n}{3} = \frac{n(n-1)(n-2)}{3!}\]
The pattern: \(\binom{n}{k}\) has exactly \(k\) factors on top (starting at \(n\) and counting down) and \(k!\) on the bottom. This form will be critical when we plug in \(n = k\) and take \(k \to \infty\).
Building Toward the Limit: Applying Binomial Expansion to \((1 + r/k)^k\)
Now we connect everything. Set \(a = 1\) and \(b = r/k\) in the binomial theorem, with exponent \(k\):
\[\left(1 + \frac{r}{k}\right)^k = \sum_{i=0}^{k} \binom{k}{i} \cdot 1^{k-i} \cdot \left(\frac{r}{k}\right)^i = \sum_{i=0}^{k} \binom{k}{i} \cdot \frac{r^i}{k^i}\]
Writing out the first few terms using our pattern:
\[= 1 + \frac{k}{1} \cdot \frac{r}{k} + \frac{k(k-1)}{2!} \cdot \frac{r^2}{k^2} + \frac{k(k-1)(k-2)}{3!} \cdot \frac{r^3}{k^3} + \cdots\]
\[= 1 + r + \frac{k(k-1)}{k^2} \cdot \frac{r^2}{2!} + \frac{k(k-1)(k-2)}{k^3} \cdot \frac{r^3}{3!} + \cdots\]
Now look at each fraction like \(\frac{k(k-1)}{k^2}\). As \(k \to \infty\):
\[\frac{k(k-1)}{k^2} = \frac{k}{k} \cdot \frac{k-1}{k} = 1 \cdot \left(1 - \frac{1}{k}\right) \to 1\]
So in the limit, every such fraction becomes 1, and we get:
When you compound interest infinitely often, the result is the exponential function \(e^r\). This connects a simple banking idea to one of the most important numbers in mathematics.
\[\lim_{k \to \infty}\left(1 + \frac{r}{k}\right)^k = 1 + r + \frac{r^2}{2!} + \frac{r^3}{3!} + \cdots = \sum_{i=0}^{\infty} \frac{r^i}{i!} = e^r\]
That infinite sum is the Taylor series for \(e^r\) — and that is how compound interest leads us to the number \(e\).
Why the Derivative of \(e^x\) Is \(e^x\) Itself
This is the property that makes \(e\) truly special. Out of all possible exponential functions (\(2^x\), \(3^x\), \(10^x\), …), only \(e^x\) is its own derivative:
The function \(e^x\) is the only exponential function whose rate of change at every point equals its value at that point. This single property is what makes \(e\) the “natural” base for calculus.
\[\frac{d}{dx}(e^x) = e^x\]
No other function grows at a rate exactly equal to its current value. If you have 100 bacteria and the population is modeled by \(e^x\), the rate of growth at that moment is also 100. The bigger it gets, the faster it grows — and the growth rate is always perfectly matched to the size.
Compare \(e^x\) with its derivative — they are the same curve:
Adjust \(b\) and notice that for most bases, the derivative (dashed) is a different curve from the function (solid). But set \(b = 2.72\) (close to \(e\)) and they nearly overlap. Only at \(b = e\) exactly are they identical.
Cheat Sheet
| Concept | Key Fact |
|---|---|
| Definition of \(e\) | \(e = \lim_{k \to \infty}\left(1 + \frac{1}{k}\right)^k \approx 2.71828\) |
| General compound interest limit | \(\lim_{k \to \infty}\left(1 + \frac{r}{k}\right)^k = e^r\) |
| Derivative of \(e^x\) | \(\dfrac{d}{dx}(e^x) = e^x\) |
| Binomial Theorem | \((a+b)^n = \displaystyle\sum_{i=0}^{n}\binom{n}{i}\,a^{n-i}\,b^{i}\) |
| Binomial coefficient | \(\binom{n}{k} = \dfrac{n!}{k!(n-k)!}\) |
| Complementarity | \(\binom{n}{k} = \binom{n}{n-k}\) |
| Taylor series for \(e^r\) | \(e^r = \displaystyle\sum_{i=0}^{\infty}\dfrac{r^i}{i!} = 1 + r + \dfrac{r^2}{2!} + \dfrac{r^3}{3!} + \cdots\) |