c) Suppose you're on an island with only a solar-powered very basic calculator. Use the result from part b to approximate
September 29th 2007, 01:41 PM
We'll prove that the sequance is low bounded by
(I used AM-GM inequality).
Now, we'll prove that the sequence is descrescent. .
Suppose that .
Then, (I used the fact that the sequence is bounded).
So, the sequence is convergent. Let .
Applying the limit in the recurrency relation we have
September 29th 2007, 03:56 PM
I recognized the function . . . and "eyeballed" the answers.
b) Let and define as being the sequence:
Prove converges and find the lim.
c) Suppose you're on an island with only a solar-powered very basic calculator.
Use the result from part b to approximate
I'm old enough to remember those "very basic calculators".
They were the size of a TI-30 (but an inch thick), weighed a pound, . . used a 9-volt battery (used up quickly by those red LEDs) .. cost about $100 and did only basic arithmetic.
Back then, we learned many tricks for approximating more complex answers. . . And one of them was square roots.
Suppose we want .
Make a first approximation,
If we're very very lucky, is exact. . . Then the two factors of are equal: .
Most likely, the two factors are not equal. . . Obviously, one is too small and the other too large.
Then a better approximation is the average of these two factors.
So we will calculate: .
Hence, is a better approximation of
Then: . is an even better approximation.
And that is the source of that recursive function: .