anonymous
  • anonymous
does the series e^n.sin(4^-n) converges as n=1 to infinity
Mathematics
  • Stacey Warren - Expert brainly.com
Hey! We 've verified this expert answer for you, click below to unlock the details :)
SOLVED
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
katieb
  • katieb
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
amistre64
  • amistre64
I would assume that it converges to zero... but that is just an uneducated guess :) take sin(4^-n).... that is sin(1/(4^n)) as the "n" gets bigger and bigger, we get a bigger and bigger number in the bottom of the fraction; large bottom fractions tend to go to zero. for example: 1/10000...000000 = .000000...00001 which is a very small number. the sin(0) = 0 which whould make the rest of the stuff = 0
anonymous
  • anonymous
You should arrange the product as such:\[\frac{\sin(\frac{1}{4^n})}{e^{-n}}\]Then in the limit as n goes to infinity, you have the indeterminate form, 0/0, so you can use L'Hopital's rule. Doing this yields\[\frac{-4^{-x}\log 4 \cos \frac{1}{4^x}}{-e^{-x}}=\frac{e^x}{4^x}\log 4 \cos \frac{1}{4^x}=\left( \frac{e}{e^{\log 4}} \right)^x \log 4 \cos (\frac{1}{4^x})\]
anonymous
  • anonymous
As x tends to infinity, cos(1/4^x) tends to 1, and since e^(log 4)>e, \[\left( \frac{e}{e^{\log 4}} \right)^x \rightarrow 0\]as x tends to infinity.

Looking for something else?

Not the answer you are looking for? Search for more explanations.

More answers

anonymous
  • anonymous
So your expression tends to 0 as x tends to infinity.
anonymous
  • anonymous
thanks a lot ! :)

Looking for something else?

Not the answer you are looking for? Search for more explanations.