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Title says it all, I have an itch about series like this that seem to fall in the gray area where classical proofs that rational partial sums that converge too quickly must converge to transcendental don't seem to apply. This is such a special case, does any one know if the series

$$\sum_{i=0}^{\infty} \frac1{k^{2^i}}$$

converges to algebraic or transcendental, perhaps the answer/lack of an answer depending on the choice of integer $k > 1$?

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(What I learned reading the Wikipedia article on transcendental numbers)

If $a$ is an algebraic number in $(-1,1)$, it has been shown that any number of the form:

$$\sum_{i=0}^{\infty}a^{2^i}$$

is transcendental. Let $a=\frac{1}{k}$ to answer your question. The article attributes this result to Loxton and references the $13$th chapter of the book New Advances in Transcendence Theory by Alan Baker.

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It follows from Liouville's theorem. The partial sums converge too well.

http://www-users.math.umn.edu/~garrett/m/mfms/notes_2013-14/04b_Liouville_approx.pdf

Assume that it is algebraic of degree $n$, and then apply the theorem.

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