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Allow me to leave alone the particular equation you mention and the issue of series, and focus instead on the general idea of finding functions "in the middle" between two families of functions. There is some extremely interesting mathematics in that idea.

The essence of this part of your question is that you have two families of functions, in your case the linear functions and the exponential functions, and the first family lies below the second in the sense that every function in the lower family is eventually dominated by every function in the upper family. Because of this, it is very natural to want to understand the functions that lie between the two classes. In what circumstances and for which types of families $L$ and $U$ can we always find a function $f$ filling the gap? That is, we seek a function $f$ that eventually dominates the functions in the lower family $L$ and is eventually dominated by the functions in the upper family $U$. It is natural to consider the cases where the families are maximal in some sense, and as a special case, one might consider what happens when they are linearly ordered by eventual domination.

Much of the content of this question is present already in the case of functions $f:\mathbb{N}\to \mathbb{N}$, and indeed, it turns out that much of the fundamental phenomenon occurs already for functions $g:\mathbb{N}\to 2$, which amounts to considering the quotient $P(\omega)/Fin$, as in this MO answer.

This way of thinking is intimately connected with the phenomenon of Hausdorff gaps.

• First, if both families are countable (or are determined by a countable sub-family, which is true in your case), then it is an enjoyable exercise to show that one may always fill the gap (first proved by Hausdorff). That is, given two countable families of functions, members of the first always eventually dominated by members of the second, then there is a function filling the gap.

• Second, Hausdorff found constructed examples of families of functions that do not admit any function in the middle; these gaps cannot be filled. That is, he produced a lower family $L$ and and upper family $U$, such that every function in the lower family was eventually dominated by every function in the upper family, but there is no function just in the middle, filling the gap. His examples were unfilled gaps having uncountable order type $(\omega_1,\omega_1)$, in the sense that the both the lower and upper families are determined by an almost-increasing $\omega_1$-sequence of functions. Thus, he constructed families of functionsthat do not admit any function in the middle; these gaps cannot be filled.

• The unfillable nature of these gaps, however, admits extensive set-theoretic independence, in the sense that an unfilled gap can sometimes be filled by a function that is added by forcing, that is, by moving to a larger set-theoretic universe. At the same time, there are methods of sealing a gap, that prevent it from ever being filled in a cardinal-preserving forcing extension.

• Kunen proved that it is consistent with Martin's axiom plus $\neg CH$ that there are unfilled gaps of type $(\omega_1,c)$ and $(c,c)$, where $c$ is the continuum, and also consistent that all such gaps are filled.

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Allow me to leave alone the particular equation you mention and the issue of series, and focus instead on the general idea of finding functions "in the middle" between two families of functions. There is some extremely interesting mathematics in that idea.

The essence of this part of your question is that you have two families of functions, in your case the linear functions and the exponential functions, and the first family lies below the second in the sense that every function in the lower family is eventually dominated by every function in the upper family. Because of this, it is very natural to want to understand the functions that lie between the two classes. In what circumstances and for which types of families $L$ and $U$ can we always find a function $f$ filling the gap? That is, we seek a function $f$ that eventually dominates the functions in the lower family $L$ and is eventually dominated by the functions in the upper family $U$. It is natural to consider the cases where the families are maximal in some sense, and as a special case, one might consider what happens when they are linearly ordered by eventual domination.

Much of the content of this question is present already in the case of functions $f:\mathbb{N}\to \mathbb{N}$, and indeed, it turns out that much of the fundamental phenomenon occurs already for functions $g:\mathbb{N}\to 2$, which amounts to considering the quotient $P(\omega)/Fin$, as in this MO answer.

This way of thinking is intimately connected with the phenomenon of Hausdorff gaps.

• First, if both families are countable (or are determined by a countable sub-family, which is true in your case), then it is an enjoyable exercise to show that one may always fill the gap (first proved by Hausdorff). That is, given two countable families of functions, members of the first always eventually dominated by members of the second, then there is a function filling the gap.

• Second, Hausdorff found unfilled gaps having uncountable order type $(\omega_1,\omega_1)$, in the sense that the both the lower and upper families are determined by an almost-increasing $\omega_1$-sequence of functions. Thus, he constructed families of functions that do not admit any function in the middle; these gaps cannot be filled.

• The unfillable nature of these gaps, however, admits extensive set-theoretic independence, in the sense that an unfilled gap can sometimes be filled by a function that is added by forcing, that is, by moving to a larger set-theoretic universe. At the same time, there are methods of sealing a gap, that prevent it from ever being filled in a cardinal-preserving forcing extension.

• Kunen proved that it is consistent with Martin's axiom plus $\neg CH$ that there are unfilled gaps of type $(\omega_1,c)$ and $(c,c)$, where $c$ is the continuum, and also consistent that all such gaps are filled.