Question

sum cos(pi*n)/(pi*n)

Answer 1

it converges using the alternating series test i think??

Answer 2

yes, are you trying to find the sum ?

Answer 3

no just if the series converges absolutely, converges conditionally or diverges.

Answer 4

for absolute convergence, \(\large \sum \left|\cos(n\pi)/(n\pi)\right|\) must also converge

see if you can show that the above series doesn't converge by comparison test with harmonic series

Answer 5

by comparing to 1/n?

Answer 6

yes

Answer 7

first of all, notice that \(\large \cos(n\pi) = (-1)^n\)

Answer 8

that means \(\large |\cos(n\pi)| = 1\)

Answer 9

rest should be easy

Answer 10

i understand it, except we use webwork for are homework and it keep saying my answer is wrong

Answer 11

whats ur answer ?

Answer 12

the series converges conditionally by comparison test

Answer 13

thats not right

Answer 14

the series converges conditionally by alternating test right ?

Answer 15

*alternating series test

Answer 16

ive also tried that. its multiple choice and i have tried all options where the series converges conditionally

Answer 17

comparison test is different

Answer 18

take a screenshot and attach if psble

Answer 19

what class is this! calc2, series are one of my favorites hehe

Answer 20

yeah calc 2

Answer 21

i hear a lot harmonic series? what do you call harmonic series

Answer 22

@ganeshie8 : Here's a bonus question for you once you're done here.

Show that \(\displaystyle\sum_{n=1}^{\infty}\frac{\cos(n\pi)}{n\pi} = -\frac{\ln 2}{\pi}\). ;-)

Answer 23

@ChristopherToni any hint on how to start xD

Answer 24

Finish helping her first, then we can discuss. :p

Answer 25

Looking at your options, i think you need to check both B and D for last series

How sure are you about remaining series above ?

Answer 26

it said i had 87% before i answered that one so that's 5/6. it said it was wrong. I'm just going to email my professor.

Answer 27

@katbeck24 : @ganeshie8 showed you that the series \(\displaystyle \sum_{n=1}^{\infty}\frac{\cos(n\pi)}{n\pi}\) was convergent using alternating series test. So D is an answer. However, since it converges by alternating series test but the absolute value of your series doesn't converge (it's harmonic), it follows that the series is conditionally convergent. Thus, B is also a valid answer. From what I can see, BD is the best answer for this. Did you already try this for answer?

Answer 28

yeah it says bd is wrong.

Answer 29

Yeah...there's something wrong there since that's the only possible answer for 6....sigh. Oh the joy of online homework... XD

Answer 30

he responded already saying not to worry about it a few other students already have brought this issue up. i wish i knew that before i spent 45min on it when i should be writing an essay.. ugh

Answer 31

Well at least we now know that it wasn't you that was wrong. :-)

Answer 32

haha yeah. thanks for the help! :)

Answer 33

\[\ln (1+x) = \sum \limits_{n=1}^{\infty}\dfrac{\cos((n+1)\pi)}{n}x^{n}\]

plugin x = 1 and i guess factor out negative 1 on right hand side
@ChristopherToni would that work or do u have any other nice trick xD

Answer 34

You got me. That's exactly what I had in mind. :-P

Answer 35

XD it kinda looks working backwards by looking at taylor series of ln(1+x)

Answer 36

Well, the series that I'm familiar with was \(\displaystyle \ln(1-x) = \sum_{n=1}^{\infty}\frac{x^{n+1}}{n}\).

Thus, \(\displaystyle \ln 2 = \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}\implies -\ln 2 = \sum_{n=1}^{\infty}\frac{(-1)^n}{n} =\sum_{n=1}^{\infty}\frac{\cos(n\pi)}{n}\).

Hence, \(\displaystyle \sum_{n=1}^{\infty}\frac{\cos(n\pi)}{n\pi} = -\frac{\ln 2}{\pi}\). XD

Answer 37

interesting!
w

Answer 38

series are really something! you guys haven't answered me
what are harmonic series?

Answer 39

another related series which taylor can compute :

\[ \sum \limits_{n=1}^{\infty}\dfrac{\cos(n\pi)}{(2n+1)\pi} = \dfrac{\pi-4}{4\pi}\]

ln(1-x) may not work here though

Answer 40

hey @xapproachesinfinity harmonic series is below series :
\[1+\frac{1}{2} + \frac{1}{3} + \cdots\]

Answer 41

oh i see
thanks @ganeshie8