Question

Help Please!

Answer 1

I dont know what to do from here....
How would I know if this was Convergent or Divergent?
Can someone explain it to me in an elementary way xD
I have a midterm on this tomorrow. D:

Answer 2

@agent0smith @mathmate

Answer 3

@johnweldon1993

Answer 4

You could use l'hopitals rule to show it, but

\(\Large \lim_{x \rightarrow -\infty} x e^x = 0\)

because the e^x approaches zero much faster than x grows.

Answer 5

So all of the \(\large e^{a/3}\) terms will be zero and it'll converge.

You seem to have made some mistakes with negative signs, btw.

Answer 6

Your value is correct but it should be positive, since the mistakes with negatives and addition, in your 2nd last line.
http://www.wolframalpha.com/input/?i=integral+from+-infty+to+6+of+re%5E(r%2F3)

Answer 7

@agent0smith Doesnt the graph of \[e^x\] approach infinity? Why would it approach 0?
Because the graph of e^x looks like this...

Answer 8

Look closer:\[\Huge \lim_{x \rightarrow -\infty} x e^x = 0\]

Answer 9

Oh negative x

Answer 10

I see. Hm

Answer 11

So you really need to know the graphs to solve these problems huh?

Answer 12

No, but it helps... I didn't use it at all. I used the fact that exponentials (like e^x) grow or shrink much faster than linear functions.

But a graph might help see that.

Answer 13

You could use l'hopitals rule to find the limit algebraically.

Answer 14

Oh I see. Would ln or Log graphs grow/shrink faster than normal linear functions as well?

Answer 15

l'hospitals how? :3

Answer 16

would you bring the r down by making the exponent negative?

Answer 17

In order of growth:

logarithms (eg. in x*ln(x) the x dominates the ln(x))
linear/polynomials
exponential\[\large \lim_{x \rightarrow -\infty} x e^x = \lim_{x \rightarrow -\infty} \frac{ x }{e^{-x} }\]notice it's currently of the form \( -\infty/\infty \) so apply l'hopitals rule\[\large \lim_{x \rightarrow -\infty} \frac{ x }{e^{-x} }= \lim_{x \rightarrow -\infty} \frac{ 1 }{-e^{-x} }=\frac{ 1 }{ -e^\infty } = 0\]

Answer 18

Oh I see.
So exponentials dominate linear
while linear dominates logarithms

and
if the answer is infinity then it is diverging,
if the answer is zero, the it is converging,

right?

Answer 19

Yes and yes.

Answer 20

What if my answer is not zero?
Like for this problem, I got 2.

Is that neither Converging or Diverging then?

Answer 21

An integral converging means it approaches an actual value. Zero is just one of those values.

Answer 22

Oh so it for the problem i got 2, is also converging

Answer 23

right? :D

Answer 24

If you get a value, yes it converges.

Answer 25

Oh okay. Thanks!