Question

Determine if the series is convergent Sum[n/e^n], n,1,inf.

Answer 1

I would appreciate if you use the equation editor (latex)
\[\large \sum_{1}^{\infty} \frac{n}{e^n}\]

Is this your question ??

Answer 2

yup

Answer 3

If i used the ratio test, i think I get "1"

Answer 4

You should not get 1 from the ratio test. I got 1/e from it.

Answer 5

@bmp , could you show how to do this as I am new to this. Or just guide me how to do this?

Answer 6

From the ratio test we have: \[ \lim_{n \rightarrow \infty} |\frac{n+1}{e^{n+1}} \frac{e^n}{n}| \implies | \frac{1}{e} \lim_{n \rightarrow \infty} \frac{n+1}{n} \]

Answer 7

But the limit is equal to 1, so we are left with 1/e. Because the limit is exists and is < 1, we conclude that the series converges.

Answer 8

Typo: the limit exists*

Answer 9

@bmp Thanks a lot :D

Answer 10

For some reason, I decided to make "e" magically disappear

Answer 11

@shivam_bhalla No problem :-) I wasn't too careful with my typing tho. If something remained unclear, ask away.

Answer 12

Thannks!!

Answer 13

@bmp Just one thing. How did you get n+1/e^(n+1) in your answer. Is their any technique ??

Answer 14

it's the ratio test rules

Answer 15

It's the ratio test: you do \(\LARGE \lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} \)

Answer 16

It's the modulus of the division. If the limit exists and is < 1, the series converges. If the limit is > 1 or does not exist, the series diverges. If the limit is equal to 1, it's inconclusive.

Answer 17

u forgot something bmp, it's the absolute of that

Answer 18

Ok I got that, But just one question. If you are taking modulus then how can you arrive at a negative value??

Answer 19

I mentioned on the second post. I don't recall the Latex for || by memory, haha.

Answer 20

You can't. < 1 means that it will be between 0 and 1 or [0,1)

Answer 21

Loks like I have to clear my glasses. :P I thought of it as less than 0 :P Thanks a lot @iHelp and @bmp

Answer 22

*looks

Answer 23

np, and bmp sorry- i missed the post

Answer 24

That's fine. Had I forgot it, you would be perfectly correct and it would help shivam_bhalla. Anyway, glad to be helpful :-)