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Abstraction Mathematics help needed watch

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

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

>0, N N such that n > N, |an - 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.

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
2. 5.(a) Give a precise definition of what it means to say that (an) converges to l as n tends to infinity.
For every there exists (note the subscript shows the dependence of on ) such that for all
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 increases gets closer and closer to .

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 and so guess that the limit is
Then:
3. 7. Determine, with jusification, for which of the following the sequence (an) converges; where a limit exists determine it.
(b) an : = (2^n)×(n²)
We can see 'casually' that as so we would perhaps be justified to disprove convergence by applying the definition for as which is for every

Hence as and is not convergent.

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.
Let
Then as
That is, as and so is divergent.

Sorry, I don't know the theory behind the first question but I'm sure someone else will be able to help.
4. Anyone know the tex for a right arrow with a cross through it?
\not\to or \not\rightarrow
5. (Original post by manps)
4.(c) Is (Z4 - { [0]4}, , [1]4) a group?
2.2 = 0 (mod 4)

So the closure axiom isn't satisfied.
6. (Original post by Jonny W)
\not\to or \not\rightarrow
Thanks

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