Question

Calculus 2 question! Help w. power series please

Answer 1

Find the radius and interval of convergence of the series
\[\sum_{n=1}^{\infty}\frac{ (-3)^{n} }{ n \sqrt{n}}x^{n}\]

Answer 2

HI!!

Answer 3

make a guess ,for the radius of convergence, i bet you are right

Answer 4

sqrt(3) ??

Answer 5

if you want to do it step by step we can

Answer 6

lol \(\sqrt{3}\)

Answer 7

Hi! i tried the problem and got infinity over infinity but that was the wrong answer. the radius of convergence should be 1/3 and interval |x|<3 but idk what i have to do to get the right answer

Answer 8

we best do it step by step

Answer 9

sorry |x|<1/3

Answer 10

but yes please. thank you

Answer 11

but one thing for sure, it is more or less obvious that
\[\lim_{n\to \infty}\frac{(n+1)\sqrt{n+1}}{n\sqrt{n}}=1\] right ?

Answer 12

this is where you say "yeah that is pretty obvious"

Answer 13

um why's that obvious?

Answer 14

think about back in some pre-calc class where you were asked for the "horizontal asympote" of
\[f(x)=\frac{(x+1)\sqrt{x+1}}{x\sqrt{x}}\] what would you say?

Answer 15

do you just divide everything by n

Answer 16

you would say "the degree of the top is the same as the degree of the bottom" (even though they are not really polynomials) so the limit is the ratio of the leading coefficients, namely 1

Answer 17

okay i see now

Answer 18

you can do it the math teacher way by dividing top and bottom by \(n\) if you like, you still get 1

Answer 19

I think that we have to compute this quantity, first:
\[\Large L = \mathop {\max \lim }\limits_{n \to + \infty } \sqrt[n]{{\frac{{{3^n}}}{{n\sqrt n }}}}\]

Answer 20

so you would just multiply 3|x| by 1 and this has to be <1

Answer 21

so now all that is left is
\[\frac{3^{n+1}}{3^n}=3\]

Answer 22

@Michele_Laino i would to with the simple ratio test, not root test

Answer 23

that means you have to solve
\[|3x|<1\] which give you
\[|x|<\frac{1}{3}\] or
\[-\frac{1}{3}<x<\frac{1}{3}\]

Answer 24

it is not the root thest, here we have a power series, so we have to apply a theory slightly different @misty1212

Answer 25

okay thank you so much. so all i have left to do is check endpoints

Answer 26

test*

Answer 27

thank you misty1212 i was just stuck on the limit part!

Answer 28

then you have to do two more things
a) replace \(x\) by \(-\frac{1}{3}\) and see what we get, then
b) replace \(x\) by \(\frac{1}{3}\) and check that

Answer 29

ok good
\[\color\magenta\heartsuit\]

Answer 30

we have:
L=3
so from the theory we can write:
r=1/L=1/3

Answer 31

you stuck with something else?