1. ## Sequence Convergence

I want to find the limit of the sequence:

$\displaystyle a_n=(-1)^n\frac{n}{n+1}$

$\displaystyle \lim_{n->\infty}a_n=\lim_{n->\infty}\mid (-1)^n \frac{n}{n+1} \mid$

$\displaystyle \lim_{n->\infty}\frac{n}{n+1}$

$\displaystyle =1$

My book says the sequence converges at -1.

I want to find the limit of the sequence:

$\displaystyle a_n=(-1)^n\frac{n}{n+1}$

$\displaystyle \lim_{n->\infty}a_n=\lim_{n->\infty}\mid (-1)^n \frac{n}{n+1} \mid$
This is NOT $\displaystyle a_n$. There is no absolute value in $\displaystyle a_n$.

$\displaystyle \lim_{n->\infty}\frac{n}{n+1}$

$\displaystyle =1$

My book says the sequence converges at -1.

Then both you and your book are wrong! For n= 10000000, say, $\displaystyle a_{10000000}= (-1)^{10000000}\frac{10000000}{10000001}= 0.999999000$, approximately, very close to 1 while if n= 1000001, $\displaystyle a_{1000001}= (-1)^{1000001}\frac{1000001}{1000002}= -0.9999998$, approximately, which is very close to -1.

This sequence does not converge. It has a subsequence (n even) that converges to 1 and another (n odd) that converges to -1.