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I am trying to determine the behavior of the following series as $n\to\infty$. Let $0<\mu<1$ be fixed and for every positive integer $n\geq 1$, consider the function $f_n(t)$ of a real variable $t$ defined by the series $\sum_{k=0}^\infty\mu^k(1-\mu^kt)^n$. I want to determine how $f_n(t)$ behaves as $n\to\infty$ for $0<t<1$ (some kind of asymptotic formula).

Clearly $f_n(t)$ converges to $0$ for each $0<t<1$, but with what rate? And say, hypothetically, the rate is $O(1/n)$, then I would need to know at least what is $\lim_{n\to\infty}nf_n(t)$. I've tried several things for two weeks and I believe the rate of $O(1/n)$ is correct, but I can't find that limit. Any suggestions?

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I am trying to determine the behavior of the following series as $n\to\infty$. Let $0\lt\mu\lt 1$ 0<\mu<1$ be fixed and for every positive integer $n\geq 1$, consider the function $f_n(t)$ of a real variable $t$ defined by the series $$\sum_{k=0}^\infty\mu^k(1-\mu^kt)^n.$$ \sum_{k=0}^\infty\mu^k(1-\mu^kt)^n$. I want to determine how $f_n(t)$ behaves as $n\to\infty$ for $0\lt t\lt 1$ (some kind of asymptotic formula). Clearly $f_n(t)$ converges to $0$ for each $0\lt t\lt 1$, but with what rate? And say, hypothetically, the rate is $O(1/n)$, then I would need to know at least what is $\lim_{n\to\infty}nf_n(t)$. I've tried several things for two weeks and I believe the rate of $O(1/n)$ is correct, but I can't find that limit. Any suggestions?

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Deterimning Determining the asymptotic behavior of a series

I am trying to determine the behavior of the following series as $n\to\infty$. Let $0<\mu<1$ 0\lt\mu\lt 1$ be fixed and for every positive integer $n\geq 1$, consider the function $f_n(t)$ of a real variable $t$ defined by the series $\sum_{k=0}^\infty\mu^k(1-\mu^kt)^n$. $\sum_{k=0}^\infty\mu^k(1-\mu^kt)^n.$$ I want to determine how $f_n(t)$ behaves as $n\to\infty$ for $0 0\lt t\lt 1$ (some kind of asymptotic formula). Clearly $f_n(t)$ converges to $0$ for each $0\lt t\lt 1$, but with what rate? And say, hypothetically, the rate is $O(1/n)$, then I would need to know at least what is $\lim_{n\to\infty}nf_n(t)$. I've tried several things for two weeks and I believe the rate of $O(1/n)$ is correct, but I can't find that limit. Any suggestions?
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