Guide

1. Compute the value of pi using the following series

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How many terms are needed to obtain an accuracy of
1e-12? How accurate is
the sum of 100 terms of this
series?

2. The Fibonacci numbers are comuted according to the following
relation:

Fn = Fn-1 + Fn-2

with
F0 = F1 = 1.

a. Compute the first 10
Fibonacci numbers.
b. For the first 50 Fibonacci numbers,
compute the ratio

Fn / Fn-1

It is claimed that this ratio approaches the
value of the golden mean
( (1 + sqrt(5))/2
). What do your results show?

3. The
Legendre polynomials (Pn(x)) are defined by the following recurrance
relation

(n+1) Pn+1(x) - (2n+1) Pn(x) + n Pn-1(x) =
0

with P0(x) = 1, P1(x) = x and
P2(x) = (3×2 - 1)/2. Compute the next three

Legendre polynomials and plot all 6 over the interval
[-1,1].

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