Hey. Currently trying to find the limit as n goes to infinity. I know it converges because I've done this before, but my question is if it converges to -0 or positive 0. This isn't something the question is asking but for my peace of mind.

\(\lim_{x\rightarrow inf} (-7)^n/(n\sqrt{n})7^n\)

bingby
Apr 18, 2017

#1**0 **

Find the following limit:

lim_(n->∞) (-1)^n n^(-3/2)

Applying the quotient rule, write lim_(n->∞) (-1)^n/n^(3/2) as (lim_(n->∞) (-1)^n)/(lim_(n->∞) n^(3/2)):

(lim_(n->∞) (-1)^n)/(lim_(n->∞) n^(3/2))

lim_(n->∞) (-1)^n = lim_(n->∞) e^(log((-1)^n)):

lim_(n->∞) e^(log((-1)^n))/(lim_(n->∞) n^(3/2))

e^(log((-1)^n)) = exp(i n π):

(lim_(n->∞) exp(i π n))/(lim_(n->∞) n^(3/2))

lim_(n->∞) e^(i π n) = e^(lim_(n->∞) i π n):

e^(lim_(n->∞) i π n)/(lim_(n->∞) n^(3/2))

Applying the product rule, write lim_(n->∞) i π n as i π (lim_(n->∞) n):

e^(i π lim_(n->∞) n)/(lim_(n->∞) n^(3/2))

lim_(n->∞) n = ∞:

e^(i π ∞)/(lim_(n->∞) n^(3/2))

i π ∞ = i ∞:

e^(i ∞)/(lim_(n->∞) n^(3/2))

Using the power rule, write lim_(n->∞) n^(3/2) as (lim_(n->∞) n)^(3/2):

(undefined)/lim_(n->∞) n^(3/2)

lim_(n->∞) n = ∞:

(undefined)/∞^(3/2)

∞^(3/2) = ∞:

**Answer: | (undefined)/∞**

Guest Apr 18, 2017

edited by
Guest
Apr 18, 2017