Doubts Regarding Evaluation of Integral

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I was encountered following example.In that I do not understand Last part.enter image description here

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I do not understand following .
Why such adjustment has been done?enter image description here Any Help will be appreciated

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$$E(z)=-\left(\dfrac{1}{2!}+\dfrac{iz}{3!}+\dfrac{(iz)^2}{4!}+\cdots\right)$$ then $$|E(z)|\leq\dfrac{1}{2!}+\dfrac{|z|}{3!}+\dfrac{|z|^2}{4!}+\cdots\to\dfrac12$$ as $z\to0$.