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Mathematics 81 Online
OpenStudy (anonymous):

What are all the values of x for which the series converges x - (x^2/2) + (x^3/3) - (x^4/4)+...[(-1)^n+1* (x^n/n)]

OpenStudy (anonymous):

try ratio test, think it is \[|x|<1\]

OpenStudy (anonymous):

but what will be my a sub n+1

OpenStudy (anonymous):

i was pretty sure it was from -1 to 1 but im not sure how 2 prove it or which of the 2 are included in the interval of convergence

OpenStudy (anonymous):

you have \[sum\frac{(-1)^{k+1}x^k}{k}\]

OpenStudy (anonymous):

so ratio is \[\frac{k}{k+1}\] limit is 1

OpenStudy (anonymous):

therefore radius of convergence is \[|x|<1\] want to try at the endpoints?

OpenStudy (anonymous):

yes

OpenStudy (anonymous):

one will work and the other will not, depending on whether you get an alternating sum or not

OpenStudy (anonymous):

so at which one is it divergent

OpenStudy (anonymous):

satellite could you help me with that one probelm again i had a question :0

OpenStudy (anonymous):

it will converge is x = 1 because you just get \[\sum\frac{(-1)^{k+1}}{k}\] an alternating sequence where the terms go to zero, so it will converge

OpenStudy (anonymous):

but at minus one the series will not alternate

OpenStudy (anonymous):

um b4: \[\frac{k}{k+1}\] limit is 1 how did u get that

OpenStudy (anonymous):

limit as k goes to infinity? it is clearly 1 right?

OpenStudy (anonymous):

hold on let me think before i type

OpenStudy (anonymous):

ok it is right,what i wrote

OpenStudy (anonymous):

if you replace x by -1 it will not converge

OpenStudy (anonymous):

why is that and which series test proves the divergence

OpenStudy (anonymous):

\[\sum\frac{(-1)^{k+1}(-1)^k}{k}\] \[\sum\frac{(-1)^{2k+1}}{k}\] \[\sum\frac{-1}{k}\] does not converge \[2k+1\] is odd and therefore \[(-1)^{2k+1}=-1\]

OpenStudy (anonymous):

\[\sum\frac{-1}{k}=-\sum\frac{1}{k}\]diverges because it is the harmonic series

OpenStudy (anonymous):

so the negative can just be taken out?

OpenStudy (anonymous):

sure why not? it is a constant

OpenStudy (anonymous):

ohhh ok i see. Thanks once again!!!

OpenStudy (anonymous):

yw

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