Question

Determine whther the series is convergent or divergent. I solved it, But I got a different answer.

Answer 1

\[\sum_{i=1}^{\infty}=\frac{ \ln n }{n ^{3} }\]

Answer 2

that would converge

Answer 3

at a certain point, the denominator would be so large, that it would almost be like adding zero

Answer 4

Nope, this is divergent, because your index is i, not n :))

Answer 5

But in all seriousness, if you let me fix it for you...
\[\Large \sum_{n=1}^\infty\frac{\ln(n)}{n^3}\]

Answer 6

@PeterPan Thanks...
I know that it is convergent...
WHat value did you get?

Answer 7

What value?
I didn't know you were supposed to evaluate the sum itself... that's so hard D:

Answer 8

well we have to...
I got a different answer from the text book...
I just wan to see if I got the value right...

Answer 9

How did you get a sum, then?

Answer 10

integrate
then find the limit

Answer 11

Integrate?!
But I thought this was a sum?
You mean like this?
\[\Large \int\limits_1^\infty \frac{\ln(x)}{x^3}dx\]

Answer 12

yes...but the initial value of x must be 2.

Answer 13

Why 2?

Answer 14

It starts with 1, doesn't it?

Answer 15

because that is where the series is decreasing

Answer 16

@whpalmer4

Answer 17

But you can very well integrate it from 1, can't you?
\[\Large = \int\limits_0^\infty ue^{-2u}du\]

Answer 18

\[\Large =\left. -\frac{e^{-2x}(2x+1)}4\right]_0^\infty\]

Answer 19

Which is \(\Large \frac34\) I believe...

Answer 20

why did you start at zero?

Answer 21

Oh, because I kind of let u = ln(x)
So, you start u from ln(1) which is zero.

Answer 22

ok...thanks..
it is 1/4

Answer 23

1/4 is the answer in the book?

Answer 24

yes

Answer 25

oops... yeah, it is 1/4
my mistake, 2(0) + 1 is most definitely not 3/4

See, this is why people shouldn't count on me ^.^

Answer 26

aww.... :<
Because you are Peter pan...

Answer 27

I make these annoying arithmetic error after an awesome calculus demonstration...

still... I AM awesome >:)

Answer 28

\[\Large =\left. -\frac{e^{-2x}(2x+1)}4\right]_0^\infty= \frac{e^{-2(0)}[2(0)+1]}{4}=\frac14\]

Answer 29

My actual name. Pan (Korean: 판) is my middle name ^.^

Answer 30

A quarter Korean, the rest is British.

Answer 31

nice...pm?

Answer 32

pm?

Answer 33

private message b.c. this is personal...
Just learned that shortcut when I was here