Find the radius and the interval of convergence sigma (X^k)/(k+1)

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Find the radius and the interval of convergence sigma (X^k)/(k+1)

Mathematics
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first take the ratio, and work the limit
\[\sum_{k=0}^{\infty} \frac{ X^k }{ k+1 }\]
I did that well. I am having troubles testing the limits

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hmm, show your work, lets see how well you got to that point then
I dont understand how to test if it decreasing in the alternating series test
so your trying to test for x=-1 ?
give me a second
\[\lim_{k\to\infty}\frac{x^{n+1}/(k+2)}{x^n/(k+1)}\] \[\lim_{k\to\infty}x\frac{k+1}{k+2}\] \[|x|\lim_{k\to\infty}\frac{k+1}{k+2}=|x|\]
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when x=1, this is a harmonic series ... when x=-1, its an alternating harmonic series ... do you recall the convergence of those?
harmonic no
how do you do it?
the harmonic does not converge, the alternating does ...
but why? Could you explain?
what is a harmonic series? and how do i know if it converges or diverges?
harmonic is\[\frac11+\frac12+\frac13+\frac14+\frac15+...\] the proofing is textbook, but it eludes me
so is like \[\frac{ 1 }{ K! }\]
http://web.williams.edu/Mathematics/lg5/harmonic.pdf
ohh my bad \[\frac{ 1 }{ k }\]
ohh ok i see if it has a K on the bottom
@amistre "the proofing is textbook, but it eludes me" 1/3 + 1/4 > 1/2 1/5 + 1/6 + 1/7 + 1/8 > 1/2 1/9 + ... 1/16 > 1/2 etc.
I still dont get how can someone recognize that a function is a harmonic series just by looking at it
the alternating harmonic series converges :) that proof eludes me as well ... just recall it from the classes.
is because of the K?
1/n is the harmonic series by definition
yes
so K by itself would be a harmonic series?
2K +1
1/k^p converges for p > 1
10 k -2
isnt it 1/k^p a p-series?
yes
ok sooo if it has a K is a harmonic series and if it has an exponent is a p series?
1/k = 1/k^1 by definition, 1/k is the harmonic series.
that is why the p series test says that if p > 1 it converges, but if 0
so it diverges?
why do i feel like this is just going in circles ....
and please special help in the alternating series one !
sorry i am just reinforcing the knowledge you are giving me by enphasing what you say lol
the alternating harmonic series converges, by properties that i cant recall either. they are explained better by the texts
http://www.math.com/tables/expansion/tests.htm
radius = 1 interval is: [-1,1)
Ohh ok. I am so sleepy right now haha. I will analyse it more tomorrow :). Thanks a lot ! anything I will send you more messages :D
im getting tired as well, but there are plenty of smarter people then me about :)

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