# Proving in the formal way that a sequence is divergent

I have the following sequence (I've already put it in the $a_n - L$ form)
$$\left|\frac{(-1)^n n+1}{n + 2}-L\right|\ge\epsilon$$
I think that in order to prove that the sequence is divergent I need to prove that $\ge\epsilon$ instead of the usual $<\epsilon$
My problem is that I have L which is stuck there so Im not sure if Im supposed to make it go somehow or play along with it like its just a number which uglify my expression

• this sequence converges. – Mustafa Said Nov 15 '14 at 19:53
• @mvw I fixed the right bar – The One Nov 15 '14 at 19:55
• @mw This sequence converges (to 1), did you mean $a_n = (-1)^n + (n+1)/(n+2)$ which diverges? – Nigel Overmars Nov 15 '14 at 19:57
• @NigelOvermars fixed the typo guys sorry – The One Nov 15 '14 at 19:57
• The divergence is of the oscillating type. You are not able to make the difference arbitrary large (as opposite to arbitrary small}. But you could show that it has two limit points. – mvw Nov 15 '14 at 20:03

Show that for all $\epsilon_{1,2}>0$ there are $N_{1,2}\in \mathbb N$ such that $|a_{2n}-1|<\epsilon_1$ and $|a_{2n+1}+1|<\epsilon_1$ for all $n>N_1,N_2$

Hence the sequence is divergent

That's a formal way but of course I started different as you... still okay for you?

• I think it is formal enough it just means that in order to prove divergence I need to prove that there are at least 2 differnt $\epsilon$ that apply to what you said? I cant find a clear definition of divergent sequence – The One Nov 15 '14 at 20:16
• After you proved the existance of the two limit points, take $\epsilon_1=\epsilon_2=0.5$ and see whats happens – Marm Nov 15 '14 at 20:20
• Divergent sequence in English just seems to mean "not convergent sequence". In German there is a distinction between "bestimmt divergent" (to $-\infty$, $+\infty$) and "unbestimmt divergent" (your case), which I can not find in English so far. – mvw Nov 15 '14 at 20:32
• @mvw I can't even find the mathematical definition (besides the not converge defintion) – The One Nov 15 '14 at 20:41
• A sequence with does not converge is called a divergent sequence :) So, just negate the definition of convergent sequence and you will get the solution together with my comment above. – Marm Nov 15 '14 at 20:44

$$\frac{(-1)^n n+1}{n + 2} = \frac{(-1)^n}{1+2/n} + \frac{1}{n + 2}$$

The subsequence of odd indices converges to $-1$. The subsequence of even indices converges to $+1$.

For the rest, read this.