and hence

Thus

i.e.

Since

it follows from the squeezing rule that

Let , and let . Let

By (6.82), we have

so

It follows from (6.83) that

i.e.

Hence

By (6.86), we have

so by the squeezing rule, , i.e.

This completes the proof of (6.85).

The reason we assumed to be positive in the previous example was to guarantee that has a logarithm. We could extend this proof to work for arbitrary , but we suggest an alternate proof for negative in exercise 6.97

I wrote a Maple procedure to calculate by using this fact. The procedure

`limcalc(n)`

below calculates
and I have printed out the results for n = 1,2,...,6.

`> limcalc := n -> (1+ .01^n)^(100^n);`

`> limcalc(1);`

`> limcalc(2);`

`> limcalc(3);`

`> limcalc(4);`

`> limcalc(5);`

`> limcalc(6);`

Explain what has gone wrong. What can I conclude about the value of from my program?

`evalf`

returns the decimal
approximation of its argument.
`> limit( (1+1/n)^n,n=infinity);`

`> evalf(%);`

Suppose that dollars is invested at % annual interest, compounded times a year. The value of the investment at any time is calculated as follows:

Let
, and let be the value of the investment at
time Then

(6.92) |

and in general

(6.93) |

For example, if denotes the value of one dollar invested for one year at % annual rate of interest with the interest compounded times a year, then

Thus it follows from our calculation that if one dollar is invested
for one year at % annual rate of interest, with the interest compounded
``infinitely often'' or ``continuously'', then the value of the investment
at the end of the year will be

If the rate of interest is 100%, then the value of the investment is dollars, and the investor should expect to get $2.71 from the bank.

This example was considered by Jacob Bernoulli in 1685. Bernoulli was able to show that .[8, pp94-97]

- a)
- Use the formula for a finite geometric series,

to show that

- b)
- Let
Use inequality (6.96) to show that

for all such that . - c)
- Prove that for all .

(Hence we have .)

Hint: Note that for all real numbers .