Qfyilyi

This week, I learned a bit more about limits, convergence and divergence.
I was given a sum of two sequences and asked to tell whether or not it is convergent, and what its limit is:

$a_n := (-1)^n + frac{1}{n^2 +1}$

which I re-wrote into

$lim_{nto infty}(-1)^n +lim_{nto infty}frac{1}{n^2 +1}$

I noticed that $lim_{nto infty}(-1)^n$ isn't convergent, whereas the latter is convergent and has the limit of $0$. That is why I'm not entirely sure whether $a_n$ is convergent or not, and got confused.

I hope someone can clear my doubts and explain their answer to me!
Thank you.

Let $u_n=(-1)^n$ and
$v_n=frac{1}{n^2+1}$.

$(u_n)$ is divergent since
$$lim u_{2n}ne lim u_{2n+1}$$

$(v_n)$ is convergent since

$$lim v_n=0.$$

the sum of a convergent sequence and a divergent one is DIVERGENT.

You should look at the global situation rather than focusing on the specific example.

The sum of a convergent sequence and a divergent one diverges. If this wasn't the case the difference of the sum and the convergent sequence would converge. That can't be as this is equal to the divergent sequence.

Let consider

• n odd $implies a_n := (-1)^n + frac{1}{n^2 +1} to -1$




• n even $implies a_n := (-1)^n + frac{1}{n^2 +1} to 1$

what can we conclude, recalling that for a convergent sequence all the subsequence need to converge to the same limit?