15 EVALUATING eIN THE previous lesson, we saw the following definition of e:
closer and closer to a decimal value for e.
2.7169 is an approximate value for e. As a more efficient approach, we can derive a sequence that
theorem (Topic 24 of Precalculus): (a + b^{n} = a^{n} + na^{n − 1}b + a^{n − 2}b^{2} + a^{n − 3}b^{3} + . . .
Now, e is the limit of that sum as n becomes infinite. When that happens, each fraction that depends on n approaches 1 because 1 is the quotient of the leading coefficients. (Lesson 4.) Therefore, on taking the limit of that sum as n becomes infinite: 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:
And so after only 10 terms, we obtain a value of e accurate to 6 decimal digits. 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}b^{2}
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