hp40g+.book Page 28 Friday, December 9, 2005 1:03 AM

Solution 3

The calculator is not needed here. Simply stating that

x

 

--

 

en increases for x ∈ [0,2] is sufficient to yield the

inequality:

x

2

--

--

1 ≤ en

≤ en

Solution 4

Since g(x) is positive over [0, 2], through multiplication we get:

x

2

--

--

g(x) ≤ g(x)en

≤ g(x)en

and then, integrating:

2

 

--

 

I ≤ un ≤ enI

 

Solution 5

2

--

First find the limit of en when n → +∞ .

Note: pressing after you have selected the infinity sign from the character map places a “+” character in front of the infinity sign.

Selecting the entire

 

 

 

 

 

 

 

 

 

 

expression and pressing

 

 

 

 

yields:

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In effect, --tends to 0 as n

 

 

 

 

n

 

2

 

 

 

 

 

tends to +∞ , so

 

--

tends to

e0

= 1 as n

tends to +∞ .

e

n

 

 

 

 

 

 

 

As n tends to +∞ , un is the portion between I and a quantity that tends to I .

Hence, un converges, and its limit is I .

We have therefore shown that: L = I = 4 – ln2

16-28

 

 

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