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I don't know how to start to determine whether $(1+i/n)^n$ is convergent as $n$ tends to infinity. I feel like I will have to use Archimedes's property but the use of $i$ is making it hard for me.

Ѕᴀᴀᴅ
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M. Calculator
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2 Answers2

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Hint:

$$\lim_{n\to \infty} \left(1+\frac{x}{n}\right)^n=e^x\tag{1}$$

How to apply this to your problem?

You have $a_n= \left(1+\frac{i}{n}\right)^n$.

Using $(1)$ we conclude that sequence converges to $e^i$.

Source( In case you want to read more about this limit) Limit of $(1+ x/n)^n$ when $n$ tends to infinity

Shweta Aggrawal
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  • When i first saw your comment, I thought that the $(1+i/n)^n = e^i$, but that doesn't work as $i=e^i\pi/2$. So I am not sure where to go from here. – M. Calculator Oct 06 '18 at 15:49
  • @M.Calculator I don't understand your above comment. What is the problem if $i=e^{i\pi/2}$? The limit of $(1+i/n)^n$ when n tends to infinity is is $e^i$ – Shweta Aggrawal Oct 06 '18 at 15:53
  • I am not sure what my original equation converges to, and from your first comment, I replaced the $x$ with an $i$ to get $e^i$ but that isn't what the equation converges to. I am just incredibly stuck to how to use your hint or how to progress on this question. Sorry for the poor explanation on my previous comment. – M. Calculator Oct 06 '18 at 15:57
  • @M.Calculator See the edited post – Shweta Aggrawal Oct 06 '18 at 16:05
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This is a little complicated but I will try and make everything clear without doing the i - substitution. In the first place equate ur function to y and find ln of both sidkmes. Apply ln rules at the RHS to obtain a rational function . Then use L'hopital's rule at the rig ht hand side and find the limit at the RHS.after everything u find natural exponent at both sides and now ur limit is determined.

user601144
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