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Straightforward algebra shows that (D_{infty } u) = sum _{k=0}^{infty } D_{k} u)) has the form claimed in (7).
Using these sums and the definitions of D k, some algebra shows that M 0 and M 1 have the form claimed in (3) and (4) respectively, where f_{r,q} u) = sum_{k=0}^{infty} u^{qk+r}/ qk+r)! quad quad r=0,ldots,q-1.
The term "advance fee" as used in this part is a fee, regardless of the form, claimed, demanded, charged, received, or collected by a licensee from a principal before fully completing each and every service the licensee contracted to perform, or represented would be performed.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com