just wanted to make sure I understood what I'm doing here.

  1. a. $5+(-1)^n$

Since -1 alternates between -1 and 1, then the smallest $a_n$ can be as $n->inf$ is 5-1 = 4, and the largest (sup) it can be is 5+1 = 6.

b. $5+(-2)^n$

$-2^n$ is either $\infty$ or $-\infty$, therefore the lim inf of $a_n$ = $-\infty$, and lim sup = $+\infty$.

c. $5+\frac{1}{n} \sin(n)$

sine is bounded between -1 and 1, and $\frac{1}{n} \rightarrow 0$ as $n\rightarrow \infty$, therefore, the lim sup is 5, and the lim inf is 5?

  1. Let ${a_n}$ = 1 if $n=2^k$ for some positive integer $k$, and $a_n=(1/n!)$ otherwise.

a. Find lim sup $a_n$ and lim inf $a_n$.

lim sup = 1. lim inf = 0.

b. find lim sup $\frac{|a_{n+1}|}{|a_n|}$

This comes out to be $\frac{1}{(1/n!)}$ for the lim sup, which is then $+\infty$.

c. Find lim sup $|a_n|^{1/n}$

this is just $1^{1/n}$, which is then 1?

  • $\begingroup$ You're right. You're doing well. $\endgroup$
    – Henry
    Nov 18, 2013 at 3:35
  • $\begingroup$ So all that I put are correct? These are simple questions so it's kind of embarassing if I can't even manage these simple concepts! $\endgroup$ Nov 18, 2013 at 3:38

1 Answer 1


Your conclusion about the first one is correct. Indeed, $$(-1)^{\text{even}}=\color{red}{+}1,~~(-1)^{\text{odd}}=\color{blue}{-}1$$ so $$4\leq a_n \leq6$$

enter image description here

For the second one, we know that for $|a|>1$ then $a^n$ may tend to $+\infty$ or $-\infty$.

enter image description here

About the third one, what happens when $n=1$. It really make the sequence to be the bigger value.i.e; $n=1$ then $a_n=5+\sin(1)<6$:

enter image description here

  • $\begingroup$ Nicely illustrated! +1 $\endgroup$
    – amWhy
    Nov 18, 2013 at 14:35
  • $\begingroup$ very nice! for the 3rd one, the lim sup is then 6? $\endgroup$ Nov 18, 2013 at 16:40
  • $\begingroup$ I would say that the lim sup is $5+sin(1)$ $\endgroup$
    – Brofessor
    Jan 24, 2017 at 1:59
  • $\begingroup$ @Resident hey, what did you use to make these graphs? $\endgroup$
    – segevp
    May 18, 2017 at 9:44
  • 2
    $\begingroup$ @segevp: dear friend, I cannot see the graphs! it is strage to me!!. But I think I did them by Maple. $\endgroup$
    – Mikasa
    May 18, 2017 at 13:23

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