Subsequential limit of $S_n=\left(-\frac{\sqrt{2}}{4}\right)^n$

Question

I'm having trouble identifying the subsequences, or patterns of the sequence.

I recognize the problem can be rewritten:

$S_n=\left(-\frac{\sqrt{2}}{4}\right)^n=-\frac{\sqrt{2}^n}{4^n}$

And if I had to take a guess at what I thought were my subsequences I'd say:

$-\sqrt{2}^n$ and $-\frac{1}{4^n}$

Using that I'd say the limit sup is 0 and limit inf is $-\infty$

Answer 1

you can write it as:$$S_{ n }=\left( -\frac { \sqrt { 2 } }{ 4 } \right) ^{ n }=\frac { \left( -1 \right) ^{ n } }{ \left(2\sqrt { 2 }\right)^n } $$ then give for n odd and even numbers in order to find subsequences

Answer 2

Try to show that

$$|x|<1\implies \lim_{n\to\infty} x^n=0$$