An Approach



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IN THE previous lesson, we saw the following definition of e:

Evaluating e

  On changing the variable from x to  1
, we have:
e = Evaluating e
  By letting n take on larger and larger values in  Evaluating e we can come

closer and closer to a decimal value for e.

Evaluating e  =  2.25
Evaluating e  =  2.489
Evaluating e  =  2.594
Evaluating e  =  2.6534
Evaluating e  =  2.705
Evaluating e  =  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  Evaluating e  let us apply the binomial

theorem (Topic 25 of Precalculus):

(a + b)n  =

an + nan − 1b  + Evaluating ean − 2b2  +  Evaluating ean − 3b3 + . . .

On putting a = 1 and b =  1
, we get

Evaluating e

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:

Evaluating e

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:

11st term     1.000000
12nd term (dividing by 1)   1.000000
13rd term (dividing by 2)   0.500000
14th term (dividing by 3)   0.166667
15th term (dividing by 4)   0.041667
16th term (dividing by 5)   0.008333
17th term (dividing by 6)   0.001389
18th term (dividing by 7)   0.000198
19th term (dividing by 8)   0.000025
10th term (dividing by 9)   0.000003
  Sum   2.718282

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,

Evaluating ean − 2b2

  show that, on putting a = 1 and b 1
,  the term becomes

Evaluating e

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
Do the problem yourself first!

Evaluating e

End of the lesson

The Owl

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Copyright © 2017 Lawrence Spector

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