Question

find the limit of sequence ((sin^2 n) / 5) ^n

Answer 1

Easier look of this problem : \[\ (\frac{ \sin ^{2} n }{ 5 }) ^ {n}\]

Answer 2

As it approaches what?

Answer 3

infinity

sorry I forgot that haha

Answer 4

Haha no problem. Well, first off, what's your guess and why? What do you know about the sine function?

Answer 5

sine x approach infinity should have the value between -1 and 1 right ?

Answer 6

and sine ^ 2 x should be between 0 and 1

Answer 7

and now I'm stuck :D

Answer 8

Sure, so think about the denominator. And while you do that, go ahead and think about two specific cases of the sin^2(n)

\[(\frac{ 0 }{ 5 })^n and(\frac{ 1 }{ 5 })^n\]

Answer 9

hmm..... are they indeterminate form 0^inf and 1^inf ?? I'm not sure 0^inf is simply 0 though.

Answer 10

Well, not exactly. Since in this case I have chosen the numerator to be 0 or 1 and it is no longer dependent on the variable.

So the first is really just 0 raised to a power, and it doesn't matter what power you raise 0 to, it's always 0.

The second one is (1/5)^n, so it's actually not 1^inf at all. Keep thinking and guessing though, I like it.

Answer 11

oh gawd it's not 1 ..... It's 1/5. So 1/5 raised to inf is .... zero ?

Answer 12

Yeah, so what about some other possible value sin^2(x) can have between 0 and 1? They're all fractions. And any number less than 1 raised to a power becomes a smaller number. So everything in the numerator is just oscillating between 0 and 1, but as we approach infinity, no matter what it is, it'll still be 1 or 0 really. So it's just like we're taking the limit as (1/5)^n approaches infinity, which is 0. It just sort of wobbles a little bit while we get there.

Answer 13

A graph is good: http://www.wolframalpha.com/input/?i=%28sin%5E2%28x%29%2F5%29%5Ex+from+0+to+2

in fact, I only picked the graph to show from 0 to 2 because it approaches 0 so quickly before getting really far at all.

Wolfram's a great tool to help you learn too if you don't use it to just solve all your calculus problems and cheat.

Answer 14

ok. got it all. Thank you very much !! :)

Answer 15

One way to approach this is to write

\[

\left | \frac {\sin^2(n) } 5 \right|^n\leq \frac 1 {5^n}

\]
and deduce.