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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
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3+0 Answers

 #1
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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
 #2
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How is this an answer

bingby  Apr 18, 2017
 #3
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Sorry, there was a typo!!

Guest Apr 18, 2017

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