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May 1st, 2018, 10:27 PM   #1
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nth - root test - is this a mistake?? Help

I need help with the following, but I am stuck. Is the i to the power n a typo? Please help me solve this question...


"Use the nth-root test to find the conditions on the constant a in order that the series

$\displaystyle \sum _{n=1}^{\infty \:}\left(\frac{an}{an+1}\right)^{n^2}i^n$


is absolutely convergent

Last edited by skipjack; May 2nd, 2018 at 03:37 PM.
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May 1st, 2018, 11:45 PM   #2
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We can't possibly know. You'll need to ask the person who gave you this problem.
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May 2nd, 2018, 12:12 AM   #3
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Thanks... i meant i have no idea how one would go about solving it. Can you perhaps assist?
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May 2nd, 2018, 12:23 AM   #4
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Sure, can you tell me what $a$ is and what $i$ is? Can we assume $i$ is the imaginary unit? Is $a$ complex? Real?
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May 2nd, 2018, 12:36 AM   #5
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I made a mistake - it should be the constant α not a
i is the imaginary unit yes
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May 2nd, 2018, 03:31 AM   #6
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Well, then, do you know what the "nth root test" (I believe I would just say "root test") is? If you do then what do you get when you apply it?
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May 2nd, 2018, 05:08 AM   #7
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Yes, here is what I have so far

$\displaystyle \:\lim _{n\to \infty }\:\sqrt[n]{\left(\frac{αn}{αn+1}\right)^{n^2}i^n}$

which, after simplifying, leaves me with

$\displaystyle \:\lim _{n\to \infty }\:\left(\frac{αn}{αn+1}\right)^ni$

To simplify I though of multiplying with $\displaystyle \frac{\frac{1}{αn}}{\frac{1}{αn}}$ which is equal to 1
Then I would be left with

$\displaystyle \lim _{n\to \infty }\left(\frac{1}{1+\frac{1}{n}}\right)^n\:i$

So then because $\displaystyle \lim _{n\to \infty }\left(1+\frac{1}{n}\right)^n\:\:=\:e$
I am left with
$\displaystyle \lim _{n\to \infty }\left(\frac{1}{e}\right)i$

I don't know if I am allowed doing that
and I don't know how to go further to "find the conditions on constant $\displaystyle α$" in order that the series is absolutely convergent''
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May 2nd, 2018, 05:14 AM   #8
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You take the nth root of the modulus in the nth root test. You forgot the modulus.
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May 2nd, 2018, 06:33 AM   #9
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What happened to $\alpha$?
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May 2nd, 2018, 08:12 AM   #10
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so will it be right to take the complex number as

$\displaystyle 0+\left(\frac{1}{e}\right)i$

and then use the formula

$\displaystyle \sqrt{a^2+b^2}$ ?


Please help... I'm more lost now than before
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