Page 203 - 'Blast_Into_Math

Blast into Math! Analatic nummer theora: ants, ghosts and giants

√ √ √
3. * Let x 1 = 1, x 2 = 1+ 1, x 3 = 1+ 1+ 1 and in general, define
√
x n+1 = 1+ x n .

Prove that this sequence converges and find its limit.

4. Let

∞
{x n } n=1

be a sequence of numbers such that the series

∞

x n
n=1

converges. Prove that

lim x n =0.
n→∞

5. Prove that, for any prime numbers q and p , the p root of q is irrational.
th
6. Prove that the complex numbers are closed under addition, subtraction, and multiplication.
Prove that every complex number has an additive inverse, and every non-zero complex
number has a multiplicative inverse.
7. A magic square is an n × n array of integers so that the sum of any row or any column is
always the same number. Make a 2 × 2 magic square, a 3 × 3 magic square and a 4 × 4
magic square. Then find a general rule for making magic squares of any size. Have fun,

because there are many different ways to do this – enjoy your mathematical creativity!
8. * Pove that for each x ∈ R , the following series converges

∞ n
x

.
n!
n=0
If you continue your analysis studies, you will be able to prove that the function

∞ x n

x
e = .
n!
n=0
If you go on to learn more analysis, you will be able to prove that the trigonometric function

sine is equal to

∞ 2n+1
x
n
sin(x)= (−1) ,
(2n +1)!
n=0

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