Abstraction Mathematics help needed

Hi, i need help with the following questions but i need the answers explained if possible.

4.(c) Is (Z4 - { [0]4},

\odot

, [1]4) a group?

5.(a) Give a precise definition of what it means to say that (an) converges to l as n tends to infinity.

The answer is below,

\forall

\epsilon

>0,

\exists

\varepsilon

N such that

\forall

n > N, |an - l| <

\epsilon

N = natural numbers
(b) By directly applying the definition, prove that,
an : = (√n)/(1-3√n)
converges to a limit, that you are expected to determine.

7. Determine, with jusification, for which of the following the sequence (an) converges; where a limit exists determine it.
(b) an : = (2^n)×(n²)

8. Determine with justification, which of the following series converge.
(c) ∑ n=1 to n=∞ (n/n+1)
You may appeal to the general limit theorems.

Thanks a lot

Reply 1

Gaz031

5.(a) Give a precise definition of what it means to say that (an) converges to l as n tends to infinity.

For every

\epsilon >0

there exists

N_{\epsilon}

(note the subscript

\epsilon

shows the dependence of

N

\epsilon

) such that

|a_{n}-l|<\epsilon

for all

n>N

This is a definition and probably just needs to be learnt. You can see from the definition that it's kind of obvious though - as

N_{\epsilon}

increases

a_{n}

gets closer and closer to

l

N = natural numbers
(b) By directly applying the definition, prove that,
an : = (√n)/(1-3√n)
converges to a limit, that you are expected to determine.

Before applying the definition we need an idea of what the limit is.
We can informally write

a_{n}=\frac{1}{\frac{1}{\sqrt{n}}-3}

and so guess that the limit is

\frac{-1}{3}

Then:

Unparseable latex formula:

|a_{n}+\frac{1}{3}|=|\frac{\sqrt{n}}{1-3\sqrt{n}}+\frac{1}{3}| \\[br]=|\frac{3\sqrt{n}+1-3\sqrt{n}}{3(1-3\sqrt{n})}| \\[br]=|\frac{1}{3(1-3\sqrt{n})}| \\[br]<|\frac{1}{1-3\sqrt{n}}| \\[br]<\frac{1}{\sqrt{n}} (n>1)\\[br]<\frac{1}{n} \\[br]<\epsilon \forall n>N=\frac{1}{\epsilon}

Reply 2

Gaz031