Let ℤ be the set of all integers and let, (20) 0 = { ∈ ℤ| = 4, for some integer }, 1 = { ∈ ℤ| = 4 + 1, for some integer }, 2 = { ∈ ℤ| = 4 + 2, for some integer }, 3 = { ∈ ℤ| = 4 + 3, for some integer }. Is {0, 1, 2, 3 } a partition of ℤ? Explain your answer.

is a partition of Z

Step-by-step explanation:

Given

for some integer k

for some integer k},

for some integer k},

and

for some integer k}.

Required

Is a partition of Z

Let

So:

So, we have:

Hence:

is a partition of Z

geometric series converge the common ratio r has magnitude less than one.

which is equivalent to

for the example we need

|2-x| < 1

-1 < 2-x   and 2-x < 1

x < 3 and x > 1

we can write that

answer: 1 < x < 3

1a. common ratio r=250/25=10 > 1     diverges

1b.   r=50/100=1/2 converges

when these converge they converge to

where a is the first term.

answer: 200

1c. r=-1   so it's not the case that |r| < 1.     diverges

2a. x is the common ratio so we converge when |x|< 1

2b.   r=-10x / -2 = 5x.

we converge when

|5x| < 1

|x| < 1/5

2c.   r=3-2x

-1 < 3-2x and 3-2x < 1

-4 < -2x and 2 < 2x

x < 2 and 1 < x

1 < x < 2

2d. [/tex] r=-48x^3/12 = 4x^3[/tex]

|4x^3| < 1

|x^3| < \frac 1 4

|x| < 4^{-1/3}