Old problem confirm \[\sum_{n=1}^\infty \dfrac{z^{n^2}}{n}=\sum_{k=1}^\infty a_kz^k\] Then \(a_k =?? \) Please, help

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Old problem confirm \[\sum_{n=1}^\infty \dfrac{z^{n^2}}{n}=\sum_{k=1}^\infty a_kz^k\] Then \(a_k =?? \) Please, help

Mathematics
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If you're replacing \(n^2\) with \(k\), that means you have this pattern: \[\begin{array}{c|cccc} n&1&2&3&4&5&\cdots\\ \hline k&1&4&9&16&25&\cdots \end{array}\] So while \(a_n\) is defined for all positive integers \(n\), \(a_k\) is only defined for perfect squares. In other words, you can transform \(a_n\) to \[a_k=\begin{cases}a_{\sqrt n}&\text{ for }n\in\{1,4,9,16,25,\ldots\}\\0&\text{otherwise}\end{cases}\]
|dw:1443666750718:dw|

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Since I have to find\( lim sup \sqrt[k] a_k\), hence a_k must be clear.
Yes, our definitions of \(a_k\) are identical.
Notice \[a_n=\frac{1}{n}~~\implies~~a_{\sqrt n}=\frac{1}{\sqrt n}\] If \(n\) is a perfect square, i.e. \(n^2=k\), then \(a_k=a_{\sqrt n}=\dfrac{1}{\sqrt{n}}=\dfrac{1}{k}\).
Sorry, swap \(n\) and \(k\) in the last set of equations :)
I interpret it like this \(\sum_{n=1}^\infty \dfrac{z^{n^2}}{n}= z + \dfrac{z^4}{2}+\dfrac{z^9}{3}+...\) compare to \(\sum_{k=1}^\infty a_kz^k = a_1z +a_2z^2 +a_3z^3 +a_4z^4+....+a_9z^9+...\) hence the equivalent part is a_1 =1 a_4 = 1/2 a_9 = 1/3 all other a_j =0
a_1 is k =1 when n=1 n^2 =1 a_4 = 1/2 when n =2, n^2 =4, hence k = 1/ sqrt n
But I am not sure about the terminology I should put there.
if \(a_k = \dfrac{1}{\sqrt k}\) , then \(limsup\sqrt[k]a_k= limsup\dfrac{1}{\sqrt[2k]k}\) and by squeeze theorem it is =1 still but if \(a_k = 1/k, \) we still have limsup =1 that is why I need confirm.
ok so i did this - \[\sum_{n=1}^{\infty}\frac{ z^\left( n^2 \right) }{ n }=\sum_{k=1}^{\infty}a _{k}z^k\] \[d \frac{ \sum_{n=1}^{\infty} z^\left( n^2 \right)n^\left( -1 \right)}{ dz}=d \frac{ \sum_{k=1}^{\infty}a _{k}z^k }{ dz}\] \[\sum_{n=1}^{\infty}\frac{ n^2z^\left( (n+1)(n-1) \right) }{ n }=\sum_{k=1}^{\infty}a _{k}\times k \times z^{k-1}\] \[\sum_{n=1}^{\infty}z ^{n+1}=\sum_{k=1}^{\infty}a_{k}\] ?
And you know that when we turn it to k, everything must be calculated w.r.t k, not n anymore. n is there to show the relationship between the original one to the new one
@imqwerty what are you doing?
i differentiated the summation on both sides of the equation
why do we have to do that?
i did that so as to remove/factor out the n^2 from the z
No need, but I think I got it, just turn the original one like this \[\dfrac{z^{n^2}}{\sqrt{n^2}}\] hence to k, it will be \(\dfrac{z^k}{\sqrt{k}}\)
Am I right?
Hence \( a_k = 1/\sqrt k\) if k = n^2
yes :)
Thank you so much, both you @SithsAndGiggles and @imqwerty . I do appreciate:)
np :) i like ur questions but can we get the same result using my method? :/
I don't think so, since a_k is a coefficient but in your method a_k is a series of variable z.
ok thanks :)

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