1
e is a limit, and no term of the sequence equals it
Every (1 + 1/n)^n is a rational number — (1 + 1/4)^4 is exactly 625/256. e is irrational, so the sequence approaches a value it never takes, however large n gets.
Lesson · Exponential and Logarithmic Functions · Exponential functions
The Number e
Pay 100% interest once a year and a dollar becomes two. Pay it in smaller instalments more often and the year-end total climbs — but not without limit. It settles toward one irrational number, 2.718281828…, and that number is e.
(1 + 1/n)^n climbs toward e as n grows
n = 1 2.000000 n = 12 2.613035 n = 365 2.714567
e = 1 + 1/1! + 1/2! + 1/3! + …
e ≈ 2.718281828459045 irrational: every decimal is a rounding
One limit, and the number it names
Core ideas
What this lesson establishes
e is not a new kind of object. It is a number, arrived at by a specific process, and the process is worth watching because two reasonable readings of it give two different wrong answers.
2
Two plausible readings, both wrong
1 + 1/n tends to 1, and 1 to any power is 1. Equally, n copies of 1/n add to 1, which suggests 2. Each argument freezes one part of the expression while moving the other, and they disagree — which is the signal that the move is not legal.
3
e^x is an ordinary exponential
e is between 2.718 and 2.719, so y = e^x sits between y = 2^x and y = 3^x and behaves exactly like them: increasing, positive, asymptote y = 0, and obedient to every exponent law. No algebra in this course changes because the base is e.
The material
Where e comes from
One dollar, one year, 100% interest, and the question of how often it is paid.
Lend a dollar for a year at 100%. Paid once at the end, you have 2. Paid as 50% twice, the second payment earns on the first: (1 + 1/2)² = 9/4 = 2.25. Four payments of 25% give (1 + 1/4)⁴ = 625/256 = 2.44140625 exactly. The pattern is n payments of 1/n, and the year-end total is (1 + 1/n)^n.
n (1 + 1/n)^n exact to 12 places
──────────────────────────────────────────────────────────────
1 (2/1)^1 2 2.000000000000
2 (3/2)^2 9/4 2.250000000000
3 (4/3)^3 64/27 2.370370370370
4 (5/4)^4 625/256 2.441406250000
12 (13/12)^12 2.613035290225
365 (366/365)^365 2.714567482022
1000 (1001/1000)^1000 2.716923932236
──────────────────────────────────────────────────────────────
e 2.718281828459
The totals rise, and the rises get smaller. Going from once a year to twice adds 0.25; going from daily to a thousand times a year adds about 0.0024. Compounding more often is worth less and less, and there is a ceiling.
Definition · The number e
e is the number that (1 + 1/n)^n approaches as n grows without bound. To fifteen places, e ≈ 2.718281828459045. It is irrational, so no decimal expansion of it terminates or repeats, and every printed value of e — here, on a calculator, in the lab — is a rounding of something that has no finite decimal form.
Theorem · The limit exists
The sequence (1 + 1/n)^n is increasing, and no term of it exceeds 3. An increasing sequence with a ceiling has a limit, and that limit is e. Both facts are established in analysis rather than here, and neither can be got from a table — a table shows finitely many terms and says nothing about the rest.
At n = 365 the two shortcuts give 1 and 2, while the true value is 2.714567…. Both readings are off, in opposite directions, on every single row. That is what makes the limit worth computing rather than reasoning about informally.
The lab's second computation is a check, not a second method
The lab also sums 1 + 1/1! + 1/2! + … and compares its finite partial sums with the compounding values. That factorial series is a fast independent way to approximate e, but deriving why it has the same limit belongs to analysis. This course neither asks you to use the series nor treats numerical agreement as a proof. The definition needed here remains the compounding limit above.
The function e^x
f(x) = e^x is the exponential function with base e. f(0) = 1, f(1) = e ≈ 2.718282, f(2) = e² ≈ 7.389056, and f(−1) = 1/e ≈ 0.367879. Every one of those decimals is rounded; the exact values are 1, e, e², 1/e, and those are the forms to carry through a calculation.
Example · The same money, two ways
1000 at 5% for 10 years, compounded once a year, is 1000·(21/20)^10 ≈ 1628.89. Compounded continuously it is 1000·e^(0.05·10) = 1000·e^(1/2) ≈ 1648.72, both rounded to the cent. The gap of about 19.83 is the whole of what continuous compounding buys over annual at this rate. “Compound Interest and Continuous Growth” derives the formula A = P·e^(rt) from the table above.
Interactive
Where e comes from
(1 + 1/n)^n, computed here rather than quoted
Method
Working with e
It is a number. Four habits keep it from being treated as anything stranger.
1
Treat e as a fixed number
It lies between 2.7182 and 2.7183. e² is a number near 7.389; it is not e squared in some special sense, and no exponent law changes because the base is e.
2
Keep it exact until the last line
Write the answer as 1000·e^(1/2) and round once at the end. Rounding e to 2.72 first and then raising it to a power throws away places you cannot get back.
3
Say where you rounded
e ≈ 2.71828 is a true statement; e = 2.71828 is a false one. Every decimal here and in the lab carries the number of places it was rounded to, and yours should too.
4
Keep a finite term distinct from the limit
For every finite n, (1 + 1/n)^n is a rational approximation below e, not the value of e. Use the supplied n when a compounding term is requested; use the symbol e when the limiting value is requested.
What e is
Score 0 / 0Choose the answer you can justify.
One dollar, compounded more and more often
The first four rows are exact fractions, so they can be checked by hand. The rest are rounded, and say so.
$1 at 100% for one year, paid in n instalments of 1/n:
once (1 + 1/1)^1 = 2 $2.00
twice (1 + 1/2)^2 = 9/4 $2.25
quarterly (1 + 1/4)^4 = 625/256 $2.4414…
monthly (1 + 1/12)^12 ≈ 2.613035 $2.6130…
weekly (1 + 1/52)^52 ≈ 2.692597 $2.6925…
daily (1 + 1/365)^365 ≈ 2.714567 $2.7145…
every second n = 31 536 000 ≈ 2.71828179 $2.7182…
the ceiling e ≈ 2.71828182846 $2.7182…
differences between consecutive rows:
0.25, 0.1914…, 0.1716…, 0.0796…, 0.0220…, 0.0037…
each extra subdivision buys less than the one before it
The quarterly figure is 625/256, exactly. That is worth doing by hand once: (5/4)⁴ = 5⁴/4⁴ = 625/256, and 625/256 = 2.44140625 with no rounding at all, because the denominator is a power of two.
The last two rows agree to six decimal places, which is closer than any bank cares about, and that is the practical content of e: past a certain frequency, compounding more often changes the answer in places nobody is counting. What it does buy is a formula that no longer mentions n at all.
For a faded limit comparison, take one dollar at 100% split into ten equal compounding periods. The first line is supplied: the final amount is (1 + 1/10)^10 = (11/10)^10. Evaluate that finite term, place it between the quarterly value and e, and state why neither the decimal nor the exact fraction equals e. Use inequalities, not matching displayed digits, as the check.
Common mistakes
1
Writing e = 2.7182.718 is e to three places, and the difference surfaces quickly: 2.718^10 ≈ 22003.6 against the true e^10 ≈ 22026.5. Round at the end of a calculation, not at the start of one.
2
Arguing the limit is 1 because the base tends to 1The base tends to 1 and the exponent grows without bound at the same time. Taking the limit of one while the other is held fixed gives 1 in one order and 2 in the other; two answers from one expression means the method is broken, not that the answer is ambiguous.
3
Treating a term of the sequence as the value(1 + 1/1000000)^1000000 is a rational number correct to five decimal places and is not e. e is what the sequence approaches; it is not any entry in the table.
Check your understanding
✓
Finish when e is just a number to you, with a known origin.
You should be able to say in one sentence where e comes from, give it to five places, call it irrational without hedging, and use e^x without pausing over the base. If e still reads as a special symbol rather than a number near 2.718, the compounding table is the thing to redo by hand.
Note: “Common and Natural Logarithms” gives base e its own notation, ln, and “Compound Interest and Continuous Growth” derives A = P·e^(rt) from exactly the table above. Why e and not 2 or 10 is the natural base for calculus is a question this course deliberately does not answer; compounding is an honest and complete reason for algebra.