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I want to show that $\prod_{n=1}^\infty (1-\frac{z}{n!})$ is convergent (or uniformly convergent) (z is complex)

Can I use the Theorem:

The infinite product $\prod_{n=1}^{\infty} (1+a_n)$ converges if and only if the series $\sum_{n=1}^{\infty} a_n$ converges.

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    $\begingroup$ That's exactly what you need $\endgroup$
    – user134824
    Commented Jan 13, 2015 at 23:05
  • $\begingroup$ @user134824 But I think, $a_n$ is real in theorem but I have complex series.. I confused.. $\endgroup$ Commented Jan 13, 2015 at 23:09
  • $\begingroup$ The theorem you are looking at may only be stated for real numbers, but it is true for complex numbers as well $\endgroup$
    – user134824
    Commented Jan 13, 2015 at 23:11

1 Answer 1

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Note that \begin{align} \lim_{N\to\infty}\left|\prod_{n=1}^{N}\left( 1-\frac{z}{n!}\right)\right| = & \lim_{N\to\infty}\exp\left(\log\left(\prod_{n=1}^{N}\left| 1-\frac{z}{n!}\right|\right)\right) \\ = & \lim_{N\to\infty}\exp\left(\sum_{n=1}^{N}\log\left| 1-\frac{z}{n!}\right|\right) \\ \leq & \lim_{N\to\infty}\exp\left(\sum_{n=1}^{N}\log\left[1+\frac{|z|}{n!}\right]\right) \\ \end{align} Due to inequality $\log (x)<x-1$ for all $x>0$, \begin{align} \lim_{N\to\infty}\left|\prod_{n=1}^{N}\left( 1-\frac{z}{n!}\right)\right| \leq & \lim_{N\to\infty}\exp\left(\sum_{n=1}^{N}\left[\frac{|z|}{n!}\right]\right) \\ = & \lim_{N\to\infty}\exp\left(|z|\sum_{n=1}^{N}\left[\frac{1}{n!}\right]\right) \\= & \exp\left(|z|\lim_{N\to\infty}\sum_{n=1}^{N}\left[\frac{1}{n!}\right]\right) \\ = & \exp\left(|z|e\right) \\ \end{align}

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  • $\begingroup$ Thanks for your answer... But I want to ask one more question... What is order of this entire function (I mean this infinite product).. I dont know what is order, do you know? $\endgroup$ Commented Jan 15, 2015 at 12:06
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    $\begingroup$ @corciacandy What do you mean by 'order'? If you are talking about convergence order towards the convergence of the product is uniform or punctually you can use the M test Weiersstrass to achieve uniform convergence. $\endgroup$ Commented Jan 15, 2015 at 12:59
  • $\begingroup$ This quuestion from my homework.. But I dont understand what is order.. by the way I used M test to show uniformly convergency. Thank you :) $\endgroup$ Commented Jan 15, 2015 at 13:24

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