# Thread: Find d such that function tends to C as n tends to infinity

1. ## Find d such that function tends to C as n tends to infinity

Hi, I need help on this question. I have an answer to it but i'm not sure its right. Here is goes,

Find d such that

$\displaystyle {4^n}$/$\displaystyle ({\frac{1}{3^n}})^{-d}\rightarrow C$ as $\displaystyle n \rightarrow \infty$

Where C is a non-zero constant.

I rearranged the problem to get $\displaystyle (\frac{4}{3^d})^n$ and then if
d ≤ 1 then $\displaystyle {4^n}$/$\displaystyle ({\frac{1}{3^n}})^{-d}$ diverges as $\displaystyle n \rightarrow \infty$
d$\displaystyle \geq$2 then $\displaystyle {4^n}$/$\displaystyle ({\frac{1}{3^n}})^{-d}\rightarrow 0$ as as $\displaystyle n \rightarrow \infty$

So only value of d that converges to a non zero constant is the value of d that makes $\displaystyle {4^n}$/$\displaystyle ({\frac{1}{3^n}})^{-d}$ = 1. Is that right? so d = $\displaystyle \frac{ln4}{ln3}$?

Thanks
Katy

2. Originally Posted by harkapobi
I rearranged the problem to get $\displaystyle (\frac{4}{3^d})^n$ and then if
d ≤ 1 then $\displaystyle {4^n}$/$\displaystyle ({\frac{1}{3^n}})^{-d}$ diverges as $\displaystyle n \rightarrow \infty$
d$\displaystyle \geq$2 then $\displaystyle {4^n}$/$\displaystyle ({\frac{1}{3^n}})^{-d}\rightarrow 0$ as as $\displaystyle n \rightarrow \infty$
what happened when $\displaystyle 1 < d < 2$?

So only value of d that converges to a non zero constant is the value of d that makes $\displaystyle {4^n}$/$\displaystyle ({\frac{1}{3^n}})^{-d}$ = 1. Is that right? so d = $\displaystyle \frac{ln4}{ln3}$?
yes