A value for e

By letting n take on larger and larger values in

we can come closer

and closer to a decimal value for e.

	=  2.25
	=  2.489
	=  2.594
	=  2.6534
	=  2.705
	=  2.7169

2.7169 is an approximate value for e.

As a more efficient approach, we can derive a sequence that converges

to e more rapidly.  To

let us apply the binomial theorem (Topic 24 of

Precalculus).

(a + b)ⁿ

=  aⁿ  +  na^{n − 1}b  + a^{n − 2}b²  +  a^{n − 3}b³ + . . . .

Notice:  Each term can be derived from the previous term.  The second term follows from the first by dividing it by 1.  The next term follows by dividing by 2.  The next term, by dividing by 3.  The next, by 4.  And so on. e is the limit of the sequence of partial sums.  Here is the sum of the first 10 terms expressed as decimals:

Thus after only 10 terms, we obtain a value of e accurate to 6 decimal places.  That is an example of a rapidly converging series.

e however, like π, is an irrational number.

Problem.   In this term of the binomial theorem,

a^{n − 2}b²

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