Question

Determine whether the sequence converges or diverges. If it converges, find the limit.

Equation in comments

Answer 1

\[a_n= \frac{ (-1)^n }{ 5\sqrt{n} }\]

\[\lim_{n \rightarrow \infty} a_n = ? \]

Answer 2

What do you think? ^.^

Answer 3

I think it converges but i don't know how to do it

Answer 4

Yep, it definitely converges :)

Answer 5

how do i figure out where it converges?

Answer 6

Use a little something called... the Squeeze Theorem ^.^

Answer 7

Maybe you'll understand better when you take a look at this...
\[\Huge \frac{-1}{5\sqrt n}\le\frac{(-1)^n}{5\sqrt n}\le\frac1{5\sqrt n}\]

Answer 8

And this is in fact, true, for all \(\large n \in \mathbb{N}\)

Answer 9

so it would be 1/5?

Answer 10

No. Obviously the sequences on the left and right are both convergent, right? But to where?

Answer 11

\[\lim_{n \rightarrow \infty} \frac{ 1 }{ 5\sqrt{n} }\]

Answer 12

Yeah... so? What's the limit?

Answer 13

0?

Answer 14

That's right :)
So, whatever the limit of your sequence is, it can't be greater than 0, right?

Answer 15

right

Answer 16

Now, what's the limit of the left sequence?

Answer 17

0

Answer 18

That's right :)
So, since it was the left sequence, your original sequence cannot have a limit that is LESS an zero, right?

Answer 19

right. So it converges at 0

Answer 20

Yes. Because its limit cannot be greater than zero, and also cannot be less than zero...
It got SQUEEZED by two convergent sequences, so to speak ^.^

Answer 21

Thanks a lot!

Answer 22

NO problem ^.^