Everyone knows about the classic $$ \sum_{i=1}^{\infty} \dfrac{1}{2^i} = 1 $$ However, is there any way to find $$ \sum_{i=0}^{\infty} \dfrac{1}{2^{2^i}} = \dfrac12 + \dfrac14 + \dfrac{1}{16} + \dfrac{1}{256} + \cdots $$
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okay i edited the post – Akshaj Kadaveru Nov 27 '13 at 17:34
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Well, it's irrational for sure... I doubt there's any way to name this infinite sum that is more clear or concise than the sum itself. – Dan Shved Nov 27 '13 at 17:36
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3In fact, it's transcendental (look for Fredholm number). As such, it's unlikely to have any "nicer" expression, just as @DanShved suggested. – Peter Košinár Nov 27 '13 at 17:41
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2http://math.stackexchange.com/questions/276892/power-series-summation-formula – Mats Granvik Nov 27 '13 at 17:43
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Transcendentality follows immediately from Liouville's theorem. – MJD Nov 27 '13 at 17:54
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Why do you think it's different than a geometric series? – Michael McGovern Feb 10 '17 at 05:28
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Do you want to show it converges or are you looking for what it converges to? – mike van der naald Feb 10 '17 at 05:33
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It is easy to show the sum converges. Giving a closed form for that value is an entirely different matter. – Brevan Ellefsen Feb 10 '17 at 05:41
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1A fairly common keyword is "lacunary series", or "gap series" to google for. It has been a nice exercise (I think) in the beginning of the 20'th century to prove transcendentality of that number ... – Gottfried Helms Feb 10 '17 at 23:43
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See also: How to find $\sum_{n=0}^{\infty}\frac{1}{2^{2^n}}$?, How to sum this series to infinity: $\sum_{n=0}^{\infty} \frac1{2^{2^n}}$. – Martin Sleziak Aug 25 '17 at 12:27
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1Perhaps an interesting additional input about generalizations and especially the alternating gap/lacunary series http://go.helms-net.de/math/divers/mo/MO_Lacunaryseries.pdf – Gottfried Helms Aug 25 '17 at 19:20
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The sum mentioned by the OP, and sums like it, are in general very difficult to deal with. The OP's series is actually called the Fredholm number, and is transcendental (see the Wikipedia page for Transcendental number).
Note: This constant is also known as Kempner's number (not to be confused with the Kempner series!).
user397701
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This is the Fredholm number as pointed out in the comments, but in case you are interested in the numerical value, a quick Mathematica calculation reveals the sum is approximately $0.816422$.
JohnD
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$$\sum_{k=0}^{\infty}\frac{1}{a^{a^k}}=\sum_{k=0}^{\infty}a^{-a^k}=\sum_{k=0}^{\infty}e^{-a^k\log(a)}$$
Let's choose the seemingly simple example $a=e$ to get rid of the logarithm. We then get the sum
$$\sum_{k=0}^{\infty}e^{-e^k}$$
Now, Wolfram Alpha can't tackle it, nor can Mathematica. Given that this is a simple example of what this sort of sum looks like, I'm betting that no known, simple closed form exists. If I'm wrong, feel free to correct me though. I would be pleasantly surprised.
Brevan Ellefsen
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