upbeat jay
Let $\sum_{n=1}^\infty \frac{sin(n)}{\sqrt n}$ absolute convergange, conditional convergance or diverge.
I can use
$\sum_{n=1}^N cos(n\alpha) = \frac{sin\frac{N\alpha}{2}cos\frac{(N+1)\alpha}{2}}{sin\frac{\alpha}{2}}$
$\sum_{n=1}^N sin(n\alpha) = \frac{sin\frac{N\alpha}{2}sin\frac{(N+1)\alpha}{2}}{sin\frac{\alpha}{2}}$
gaunt trailBOT
Let $\sum_{n=1}^\infty \frac{sin(n)}{\sqrt n}$ absolute convergange, conditional...
uneven stormBOT
mtr123
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opal sphinx
first of all, the series $\sum{n=1}^\infty \frac{sin(n)}{\sqrt n}$ converges conditionally
sorry, $\sum_{n=1}^{\infty }\frac{sin(n)}{\sqrt{n}}$
uneven stormBOT
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opal sphinx
can you explain more on what you were saying about using those two series?
upbeat jay
if we take the absolute value of it we get $\sum_{n=1}^\infty \frac{|sin(n)|}{\sqrt n} \leq \sum_{n=1}^\infty \frac{1}{\sqrt{n}} \leq \sum_{n=1}^\infty\frac{1}{n}$ which diverge. so it's either diverge or conditional
uneven stormBOT
mtr123
opal sphinx
can we apply the dirichlet test for convergence for sin(n)?
opal sphinx
with form $\sum_{}^{}a_{n}b_{n}$ , where here $a_n = sin(n)$ has partial bounded sums
uneven stormBOT
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uneven stormBOT
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opal sphinx
and goes to 0 when n goes to infinity
upbeat jay
so we can apply the theorem
just how you show the sum of sin(n) is bounded?
oh ok i could use those formulas i guess?
those i posted at the beginning of the thread
opal sphinx
i think so?
upbeat jay
ok cool and also the thing i wrote about absolute convergance?
opal sphinx
i wait let me see the formulas you posted
upbeat jay
ok sure 🙂
opal sphinx
ok cool and also the thing i wrote about absolute convergance?
no it's not, it's actually convergent conditionally , just that.
and by the way you can use the fomula for the sum sin(n) if you want to
$\sum_{n=1}^N sin(n\alpha) = \frac{sin\frac{N\alpha}{2}sin\frac{(N+1)\alpha}{2}}{sin\frac{\alpha}{2}}$ this one ?
uneven stormBOT
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upbeat jay
yeah
and i mean, i can show it is not absolute since if it was we could bound it by 1/n
from below
opal sphinx
like compare it to a harmonic series?
upbeat jay
yeah
opal sphinx
well if it's that you're absolutely correct
upbeat jay
i mean
upbeat jay
huh cant use it, since sin(n) /n < 1/n so it is bounded from above and not below
opal sphinx
that's what i was saying
opal sphinx
yea
upbeat jay
so ain't it absolutely converge?
opal sphinx
$\sum_{n=1}^{\infty }\frac{\left| sin(n) \right|}{\sqrt{n}}$ diverges so how does it absolutely converge??
uneven stormBOT
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upbeat jay
how come it diverge?
a_n = |sin(n)| - bounded
b_n = 1/sqrt n goes to 0 monotincally
so we get sum anbn has limit
no?
upbeat jay
it mustn't be true, since sin(n)/sqrt n \leq 1/sqrt n which diverge
ill think about it
anyway what did you want to ask earlier?
you helped me solve the rest so i'll get it from here no problem
opal sphinx
oh i wanted to ask you how come the series diverges $\sum_{n=1}^{\infty }\frac{\left| sin(n) \right|}{\sqrt{n}}$ so how will the other one converges absolutely with the dirichlet test just that
uneven stormBOT
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upbeat jay
probably sum|sin(n)| isn't bounded
well for sure, since we only take the positive values
opal sphinx
no i think because of the nature of sin(n) it converges
opal sphinx
probably sum|sin(n)| isn't bounded
of coufrse
opal sphinx
it only takes the positive values so you can't do the dirichlet test
opal sphinx
i don't even get it i want it to be my problem too
upbeat jay
haha
upbeat jay
sure
opal sphinx
let's apply the dirichlet test,
here, a_n = sin(n)
and b_n = 1/sqrt(n)
b_n goes to 0, monotically converges when n goes to infinity
i mean b_n sorry
the partial sums of sin(n) are bounded right?
but how can we say so?
upbeat jay
i need to make sure actually but i think it will work out using that formula
opal sphinx
yes use it for more clearer proof
upbeat jay
yeah, need to think how exactly but yes
opal sphinx
oh okay when you apply it tell me what you got
upbeat jay
oh wait
opal sphinx
yea?
upbeat jay
so its whatever on the numerator which is less than 1
divided by sin(0.5)
so its bounded
thats it
upbeat jay
what do you mean? we are looking at the partial sums
so the partial sum S_N is exactly this
opal sphinx
haha
upbeat jay
thank you very much!
opal sphinx
see you later bro
upbeat jay
can't thank you?
opal sphinx
you can't?
upbeat jay
haha sorry
opal sphinx
it's worthless anyways
upbeat jay
😛
opal sphinx
XD
upbeat jay
you got many thanks from me anyway!
opal sphinx
no problem !
wide spire