
Sequences
1a) Define as being the sequence:
Prove converges and find the lim.
b) Let and define as being the sequence:
Prove converges and find the lim.
c) Suppose you're on an island with only a solarpowered very basic calculator. Use the result from part b to approximate

1) a
We'll prove that the sequance is low bounded by
(I used AMGM 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

Hello, alikation0!
I recognized the function . . . and "eyeballed" the answers.
I'm old enough to remember those "very basic calculators".
They were the size of a TI30 (but an inch thick), weighed a pound,
. . used a 9volt 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: .
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Actually, I used this form: .