how do i find wheter this is absolute or conditional convergence, can someone offer some guide & advice,thank you so much. im out of ideas for this.
i)
ii)
thank you.
always check for absolute convergence first because if a series converges absolutely you're done. If not then check for conditional convergence.
For 1) sin(npi/2) = -1,0 or1
|sin(npi/2|/n! < 1/n! and the series for 1/n! converges (easy to check with ratio test)
lim n/ln(n) = inf therefore the series does not converge absolutely
also since lim n/ln(n) = inf then series (-1)^n n/ln(n) diverges
as lim (-1)^n n/ln(n) = DNE
thank you so much for replying Calculus26.from your explanation, both equations converge conditionally, am i right?
so for sin(npi/2) = -1,0 or1, the series converges conditionally?
for [tex]
please help to clarify, i still can't understand fully. also, what does DNE mean?
sorry to trouble you, thank you so much for helping.
for
|sin(npi/2|/n! < 1/n & 1/n! converges,
the equation converge absolutely.
ii)lim n/ln(n) = inf then series (-1)^n n/ln(n) diverges
the equation converge conditionally.
is this correct? im stuck for proving with question 2, need someone to help clarify,thank you so much.
you have it correct for 1) converges abs
However 2) diverges period sinceRemeber if the sequence does not converge to 0 the series diverges.also since lim n/ln(n) = inf then series (-1)^n n/ln(n) diverges
as lim (-1)^n n/ln(n) = DNE
there are 3 possibilities -- a series can conv abs,con conditionally, or diverges
An eg of conditional convergence is the series (-1)^n/ n
The series |(-1)^n/ n| = series 1/n diverges
however series (-1)^n/ n converges by the alternating series test
lim1/n = 0
just confirmed, there is an error in question 2,n=2,just like the feedback from felper, thank you felper.
ii)
how do i prove this equation converge now,absolute or conditional., can anyone help me. thank you so much friends for helping me thus far, really appreciate.
thank you.
thank you everyone for helping, but im really confused as to why this series diverges and why .
if someone can explain this it will be very helpful. thank you so much everyone for helping, sorry to trouble all of you, im really out of ideas.
from the comments:
An eg of conditional convergence is the series (-1)^n/ n
The series |(-1)^n/ n| = series 1/n diverges
however series (-1)^n/ n converges by the alternating series test
lim1/n = 0
is it that (-1)^n/ n is taken from the original equation, just minus the ln from the equation? just wondering how we got that.
just wonderring about this few explanations, if someone could help me, it will be great. thank you so much.