#How to tell if div, cond conv, conv, or none?

59 messages · Page 1 of 1 (latest)

iron loomBOT
deft summit
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You can make use of Leibnitz Theorem because it's alternating

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cos(πn) = (-1)^n

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Your proof is basically saying nothing. You only showed that the sequence is converging towards 0 (which is a necessary condition for convergence but not a sufficient one)

proper river
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But then how will I be able to test if it is conditonally convergent too?

deft summit
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No

proper river
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Conditionally I'd have to test if an is convergent but |an| is divergent

jade irisBOT
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adonhs

587715738484211754.png
proper river
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So then I was able to prove the first part no? Because I showed lim as n approches infinity for an = 0

deft summit
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yea

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now you need the monotony part

proper river
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What is monotony? I have not heard before in lecture

deft summit
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and for absolute convergence take a look at 1/(8n-5)

deft summit
proper river
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so then show |1/8n-5| is convergent for absolute convergence?

deft summit
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no..

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Σ 1/n doesnt converge then Σ 1/(8n-5) doesnt converge as well

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So the absolute convergence fails

proper river
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oh you use comparison test?

deft summit
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For example

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  • Integral critererion
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1/n ~ 1/8n ~ 1/(8n-5) basically

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so Σ |a_n| fails

proper river
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ohh i see, so if this fails do i test conditionally convergence?

deft summit
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yes

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you only need to prove monotony

proper river
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if Σ |a_n| is divergent, now I have to see if Σ a_n is convergent?

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if sequence is increasing, what does that mean?

deft summit
proper river
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ok

deft summit
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If a seq. alternating + monotnous (increasing or decreasing) from absolute value then the series converges if the seq. converges to 0

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additionally for absolute convergence

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check if the same series but wirh absolute value converges too

proper river
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I see that the graph of the series if increasing but also oscilliating on x axis

deft summit
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which fails here due to comparison test or integral criterion

deft summit
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You consider if |a_n| is monotonous

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which means

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1/(8n-5)

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Which is essentially like 1/x

proper river
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so 1/x is always increasing

deft summit
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both decreasing starting from x = 1 = n

deft summit
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,w plot 1/x

jade irisBOT
proper river
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always decreasing

deft summit
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Here you prove it for n > 0

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Or n >= 1

proper river
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showing this will prove conditional?

jade irisBOT
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adonhs

587715738484211754.png
deft summit
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if it is not absolute, it is conditional

proper river
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ohh ok so then this series is conditionally ocnvergent?

deft summit
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yes

proper river
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ok thanks alot

left perch
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.solved