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May 1st, 2018, 10:27 PM  #1 
Newbie Joined: May 2018 From: south africa Posts: 11 Thanks: 0  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 nthroot 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. 
May 1st, 2018, 11:45 PM  #2 
Senior Member Joined: Oct 2009 Posts: 733 Thanks: 246 
We can't possibly know. You'll need to ask the person who gave you this problem.

May 2nd, 2018, 12:12 AM  #3 
Newbie Joined: May 2018 From: south africa Posts: 11 Thanks: 0 
Thanks... i meant i have no idea how one would go about solving it. Can you perhaps assist?

May 2nd, 2018, 12:23 AM  #4 
Senior Member Joined: Oct 2009 Posts: 733 Thanks: 246 
Sure, can you tell me what $a$ is and what $i$ is? Can we assume $i$ is the imaginary unit? Is $a$ complex? Real?

May 2nd, 2018, 12:36 AM  #5 
Newbie Joined: May 2018 From: south africa Posts: 11 Thanks: 0 
I made a mistake  it should be the constant α not a i is the imaginary unit yes 
May 2nd, 2018, 03:31 AM  #6 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 896 
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?

May 2nd, 2018, 05:08 AM  #7 
Newbie Joined: May 2018 From: south africa Posts: 11 Thanks: 0 
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'' 
May 2nd, 2018, 05:14 AM  #8 
Senior Member Joined: Oct 2009 Posts: 733 Thanks: 246 
You take the nth root of the modulus in the nth root test. You forgot the modulus.

May 2nd, 2018, 06:33 AM  #9 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,912 Thanks: 1110 Math Focus: Elementary mathematics and beyond 
What happened to $\alpha$?

May 2nd, 2018, 08:12 AM  #10 
Newbie Joined: May 2018 From: south africa Posts: 11 Thanks: 0 
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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mistake, nth, root, test 
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