Limit of sequence

**Len** · March 12th 2008, 02:28 PM

Using n>=N ==> | A_n - L | < e

Most the proofs I read say to start with | a_n - 1 | < e and work backwards to find N but I have done so and I cannot eliminate then n term. I am stuck and any tips / help would be hugely appreciated.

I end up getting

1+2n - (-1)^n
-------------- < e
n^2 +2n+1

**topsquark** · March 12th 2008, 02:30 PM

Originally Posted by Len

Most the proofs I read say to start with | a_n - 1 | < e and work backwards to find N but I have done so and I cannot eliminate then n term. I am stuck and any tips / help would be hugely appreciated.

I end up getting

1+2n - (-1)^n
-------------- < e
n^2 +2n+1

Your sequence definition makes no sense. The left hand side is independent of n, but you are using that to define your $a_n$ .

-Dan

**Len** · March 12th 2008, 02:32 PM

Originally Posted by topsquark

Your sequence definition makes no sense. The left hand side is independent of n, but you are using that to define your $a_n$ .

-Dan

Sorry I wrote that wrong. I am trying to show that A_n converges to 1.

**ThePerfectHacker** · March 12th 2008, 02:45 PM

You want to prove $a_n \to 1$ .

Thus, we have to show $\left| \frac{n^2+(-1)^n}{(n+1)^2} - 1 \right| < \epsilon$ .

Thus, $\left| \frac{n^2 + (-1)^n - (n+1)^2}{(n+1)^2}\right| < \epsilon$ .

Thus, $\left| \frac{n^2 + (-1)^n - n^2 - 2n - 1}{(n+1)^2} \right| < \epsilon \implies \left| \frac{(-1)^n - 2n}{(n+1)^2} \right| < \epsilon$ .

But note, $\left| \frac{(-1)^n - 2n }{(n+1)^2} \right| \leq \frac{2n+1}{n^2} \leq \frac{2n + n}{n^2} \leq \frac{3}{n}$ .

So if we can make $3/n < \epsilon$ then $|a_n - 1| < \epsilon$ . And this of course can be made by choosing $N > 3/\epsilon$ .

**Len** · March 12th 2008, 03:16 PM

Originally Posted by ThePerfectHacker

You want to prove $a_n \to 1$ .

Thus, we have to show $\left| \frac{n^2+(-1)^n}{(n+1)^2} - 1 \right| < \epsilon$ .

Thus, $\left| \frac{n^2 + (-1)^n - (n+1)^2}{(n+1)^2}\right| < \epsilon$ .

Thus, $\left| \frac{n^2 + (-1)^n - n^2 - 2n - 1}{(n+1)^2} \right| < \epsilon \implies \left| \frac{(-1)^n - 2n}{(n+1)^2} \right| < \epsilon$ .

But note, $\left| \frac{(-1)^n - 2n }{(n+1)^2} \right| \leq \frac{2n+1}{n^2} \leq \frac{2n + n}{n^2} \leq \frac{3}{n}$ .

So if we can make $3/n < \epsilon$ then $|a_n - 1| < \epsilon$ . And this of course can be made by choosing $N > 3/\epsilon$ .

Thanks for help but my last question is do i need the = between

|a_n - 1| <= 3/n . I do not think they are ever equal.

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