Infinitely Print Zeno's Dichotomy Paradox (1/(2^n))

Question

Wikipedia: Zeno's Dichotomy Paradox

An infinite number of mathematicians walk into a bar. The first one orders a beer. The second one orders half a beer. The third one orders a fourth of a beer. The bartender stops them, pours two beers and says, "You're all a bunch of idiots."

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Print the following series for as long as the program runs, with the denominator of each item being multiplied by two each time:

1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...

As n approaches infinity, the sum of this sequence approaches 2.

Rules

No, you may not print 2.

You may not print 1/1 as the first item.

You may remove spaces 1+1/2+... or add spaces 1 + 1 / 2 + ... as you need.

You may use newlines instead of spaces as a delimiter due to popular demand.

You may append a . plus a constant number of 0s to the denominator if need be.

"Infinitely" means no unnecessary delays, and for as long as possible limited by the current (variable) system's specs, but not limited by your current language.

Standard loopholes apply.

This is code-golf, so shortest answer in bytes wins.

Answer 1

05AB1E, 10 9 bytes

Saved 1 byte thanks to Erik the Outgolfer

[No…+1/J?

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Explanation

[           # loop over N infinitely [0 ...]
 No         # calculate 2^N
   …+1/J    # join with the string "+1/"
        ?   # print without newline

Answer 2

Python 2, 30 bytes

-5 thanks to Erik the Outgolfer

i=1
while 1:print i,'+1/';i*=2

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Answer 3

APL (Dyalog Unicode), 15 bytes

More fun if ⎕FR (Floating-point Representation) is 1287 (128 bit decimal) and ⎕PP (Print Precision) is 34.

{∇2×⊃⎕←⍵'+1/'}1

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{…}1 apply the following function on the number 1:

 ⎕←⍵'+1/' print the argument and the the string

 ⊃ pick the first one (i.e. the argument)

 2× double that

 ∇ tail call recursion on that (optimised, so it can be infinitely repeated)

Answer 4

Jelly, 12 bytes

‘;“+1/”ȮḢḤ’ß

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Answer 5

Pyth, 10 bytes

.V0^2b"+1/

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Answer 6

C (gcc), 60 bytes

f(){for(long long n=1;n;n*=2)printf(&"+1/%llu"[n^1?0:3],n);}

Goes up the the unsigned 64bit limit.

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This one goes on forever; (it's as small as it's going to get)

C (tcc), 312 264 255 251 233 231 208 204 195 190 188 170 bytes

l=4;c;main(i){for(char*t="*\b1+";puts(i=t),*++t=48;l=asprintf(&t,"1/%s",t+=*++t<49))for(t+=l-2;--t>i;)*t>52?*t=*t*2%58+c,c=*--t>52,*t=*t%48*2%10+49:(*t=*t*2-48+c+(c=0));}

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Here's the not so golfed version;

c;l=4;main(i){
    for(char*t="*\b1+";puts(i=t),*++t=48;l=asprintf(&t,"1/%s",t+=*++t<49))
        for(t+=l-2;--t>i;)
            *t>52?*t=*t*2%58+c,c=*--t>52,*t=*t%48*2%10+49:(*t=*t*2-48+c+(c=0));
}

Answer 7

Bash, 33 bytes

echo 1;yes|awk '{print"+1/"2^NR}'

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Change print for printf and echo for printf to avoid newline

Answer 8

dc, 19 18 bytes

1[rdp+[+1/]Prdx]dx

Explanation

We push 1 and [rdp+[+1/]Prdx] onto the stack. We then duplicate and execute [rdp+[+1/]Prdx]. The first thing it does is to rotate the stack (r) so that the value is on top. dp+ prints the value and adds itself (to multiply by 2). [+1/]P prints the invariant +1/, then we rotate the arguments so the saved macro definition is at the top, duplicate it and start again.

Notes

GNU dc will normally wrap at 70 columns. To override that and get an infinite line, add DC_LINE_LENGTH=0 to your environment variables.

Output (partial)

1
+1/2
+1/4
+1/8
+1/16
+1/32
+1/64
+1/128
+1/256
+1/512
+1/1024
+1/2048
+1/4096
+1/8192
+1/16384
+1/32768
+1/65536
+1/131072
+1/262144
+1/524288
+1/1048576
+1/2097152
+1/4194304
+1/8388608

...

+1/1042962419883256876169444192465601618458351817556959360325703910069443225478828393565899456512
+1/2085924839766513752338888384931203236916703635113918720651407820138886450957656787131798913024
+1/4171849679533027504677776769862406473833407270227837441302815640277772901915313574263597826048
+1/8343699359066055009355553539724812947666814540455674882605631280555545803830627148527195652096
+1/16687398718132110018711107079449625895333629080911349765211262561111091607661254297054391304192
+1/33374797436264220037422214158899251790667258161822699530422525122222183215322508594108782608384

...

+1/39508742960703973701699058249321147362308454847936695835596789444561799223775119383138339311531617762178533629297475770286145628088251144169939127129573351351249626456638593338094796526172345495933122070775929580836041561599159299077777828204070582743494963385663317626578994458063693594803196645315374243921729198658112825073342949389812045193177027475122956756923784327372961582710038182537333922141259552951298907303962304838080273638036760780116072004845095527233255588553482279296774697359782192465695997942318865695221244129734666070151871832662077408140671163928915923237432567622143064546106133941951721500522645450228508984172259490047633694700073310127973759870498531879627921659506889467789827772217974527938921063212560947657822971770488425182948365680848383370895735146874307161886839006536947855165776686876541793136811167263580571304093957605985666196690184928497442785360764093478017451017746734495498153095432458734678594349803378385365288877714481483657768524929803674839527329386788921377623659662146451386624096645456088162412346874407275939999997795291704727515081702280114195915760730429971004053680396194153924851183822814089628520891784451651635432882834189518985962518491913745942769608364517349644317316883786394188027676315875802300614941521167533068776637714822997427956604900326486460019184026748377784334267448781151607331163551531249017855289595742337389988164318002911475316716184901518510216506967810214179264582001641604756838614530649510787748710515970103343413387907629523554359272907613750422053342161896933352596651762204311019492086408490451558722795552457312304151493327720013539795791396780129219445263809178656123790439902010266636225369198929939010768486889471574583687740703237723884060884341313109270396042776985369907837483449193699604446325434929715621310327967316917850518470187214490004577110243561029618419720922752928800741506704602272258298159626952200574962798177352636643822057543717726388960413069933543259522625754349233326342205092206727017922426027514405659437983511440608344700472705601338936605686966987172124738772188516007048230320749819398162536814976710592597538734137705950027359014588059062707620255218965805767357796513747447844523027915137840821345660865344374642005264231547974191720135374131636753072733967472098468431960701445604771410636863217872933622336548752977495855780251562187680752211721969296325463600172622609537613191880378481131363367650069328868188689922987810873631614557359705328210279446539258010268136177047368192438059279482528102638992531928429557597690537180509886930955777942766788025773602355396907906599024775244247870956027189054457531975304478521371381521485353149305097709658931812220918262294486689602972745683542729705723801006740031632845873373515192486849474031812571748034543997685658809569510355793325451872403514779086790086080249085626688289527240523390934950310034283577812811345070405593391658025105124251311424792323428537521834833098496849926115166627987693425225164221065888040679817710055149311700218473229573882735007326736184826228276198795800430996351794893869878616496964408658079755593435250740266925743063669693033962166638727208958848108618279706541681952402424403951452727855528325863478625420852111034874224126332581600270993719540233767913017317031917301947314301381305395778148662411240099861212350963538319161122105525140542957498243160511121081190999873793463700982194274391610079035649378156907404302109291456549051145972206937941256836207058890582488706269225196655757106314491362835961785278224285066734724349363542317085673599144544341787159375484193012654526110097825033479040113441770808525499628054646905495343469874033800453592330413492501029466619548807703349840857340023090845637275878427551483951820919716838598680072870886448655557475103895987586594457323290438277217416546924072119126008341434351372116758820210364483921779586954047645736749754706129527581706364540786199925444106851308928894123838719887248427406193523873241814252112960242471157146353688570100448110981388797252642426883393630199994048973788819157093688848496407354840126057329475558680721916931454066842963283515754585176372242775627387907273143150575761016992207751724577096752997060539405477817135276919229706042957713879187001161593522722567120109358147997326512389975917858806062700348362448471895255353617508014155741101819511284391838733058694543470468138671203784913218615013226104282579286537264875430408731136448578391399523220650329513170358814645003789376544233562516359743496023216403186793980621303443495530465939251724808374274310458025502798481081725847371166851046629335315923068250771335653634693337322262107406949362155173289000790264289820372935353554150505523064644792733552895751817412405124032479006980345444326870503243741298708595698623165439665394544936179141938567338846215386320683815738357770604967341613719725594913660580596082665203735813088294781572483642298773982857912441051544424056641771245170643793444471505411285243737462083781831896533323303672072505718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7029604801221984100983880306551144728065542617917554688

...

Answer 9

GolfScript, 19 bytes

1{.'+1/'+puts.+.}do

Try it online!

Answer 10

CJam, 14 bytes

1{_o"+1/"o2*}h

Try it online!

Answer 11

><>, 14 bytes

1:n"/1+"ooo2*!

Try it online!

Answer 12

Pyth, 10 bytes

#h~hyZ"+1/

Z starts out as zero. ~hyZ post-assigns the value of 2*Z+1 to Z. Thus, Z becomes 0, 1, 3, 7, 15, ... over successive iterations. h then prints out the value one greater. # runs the infinite loop, and "+1/ gets the formatting right.

Answer 13

JavaScript (ES6), 36 34 bytes

for(a=1;;a*=2)console.log(a+'+1/')

Try it online!

Inspired by Jake Taylor's answer.

Note that this is limited by language since a is a floating-point variable, not an integer.

-2 bytes thanks to @Stefnotch.

Answer 14

Ruby, 27 25 bytes

a=1;a*=2while$><<a<<'+1/'

Try it online!

Answer 15

R, 35 34 bytes

cat(i<-1);repeat cat("+1/",i<-i*2)

Spacing is a bit werid but I understand that's ok.

Answer 16

Java, 107 102 bytes

()->{for(java.math.BigInteger z=null,o=z.ONE,n=o;;n=n.add(n))System.out.print(n.max(o)==o?1:"+1/"+n);}

z=null exists to shorten the o=java.math.BigInteger.ONE into z=null,o=z.ONE, saving 12 bytes. z.ONE will not throw a NullPointerException because we access a static member and not an instance one.

Using int shortens the code, but fails to comply after 32 iterations.

Saves

• 107 -> 102 bytes: n.compareTo(o)>0 turned into n.max(o)==o, thanks to an idea given by @Shufflepants

Answer 17

Vim, 22, 21 bytes/keystrokes

qqyiwA+1/<esc>p@"<C-a>@qqxX@q

While testing this, you might run into issues with the current register values. To fix this, run

:let @q=''
:let @"=''

before running this, or by launching vim with

vim -u NONE -i NONE

Answer 18

Befunge 93: 14 bytes

1:.2*"/1+",,,#

Answer 19

PHP, 32 Bytes

for(;;)echo bcpow(2,$i++)."+1/";

Online Version

-6 Bytes if values like 9.2233720368548E+18 are allowed

for(;;)echo 2**$i++."+1/";

Try it online!

Answer 20

Powershell, 34 bytes

for([bigint]$i=1;;$i*=2){"$i+1/"}

Try it online!

Answer 21

QBIC, 18 bytes

?1{q=q*2?@+1/`';`q

Prints each term on a new line.

Explanation:

?1       Prints 1
{        Infinite loop
q=q*2    Doubles q, starts at 1*2=2
?@+1/`  Prints the string +1/ 
  ';`      without tabs, newlines or other terminators (code literal, ; is a QBIC function)
q        Also prints q
         The infinite loop is auto-closed by QBIC at EOF.

We can save a byte with a more liberal output format:

{?q,A,┘q=q*2#+1/

Answer 22

Go, 102 100 bytes

Go can be almost as bad as Java, apparently.

import(."fmt"
."math/big")
func main(){n:=NewInt(1);for{Print(n.String()+"+1/");n.Mul(n,NewInt(2))}}

Try it online! (Would be a good idea to avoid running any of these locally. :P)

Answer 23

Aceto, 20 bytes

 pL*
pd12<
p"M"
1+1/

Prints the sequence without any spaces. When run, you won't see anything for a little while, because of buffering, run with -F to immediately see everything.

1. Pushes and prints a 1, then stores "+1/" in quick storage (the register).
2. Pushes a 1.
3. Multiplies by two, loads from the register, prints, duplicates, and prints.
4. GOTO 3.

Answer 24

JavaScript (ES6), 45 43 42 bytes

Saved 2 bytes, thanks @DanielM ! Saved 1 byte, thanks @eush77 for pointing it out.

=console.log;a=1;(1);for(;;)_(+1/${a*=2})

=console.log;(a=1);for(;;)_(+1/${a*=2})

_=console.log;for(_(a=1);;)_(`+1/${a*=2}`)

My first go at Codegolf, go easy!

Answer 25

Mathematica, 25 bytes

1//.n_:>(n~Print~"+";n/2)

Answer 26

C#, ̶6̶8̶ 154 bytes

void A(int b=1){System.Console.Write($"1{(b>1?"/"+b:"")}+");A(b*2);}

Here is a version not constrained by int

using System.Numerics;BigInteger b=new BigInteger(1);void A(){System.Console.Write($"1{(b>1?"/"+b:"")}+");b=BigInteger.Multiply(new BigInteger(2),b);A();}

Answer 27

AWK, 3732 bytes

BEGIN{for(;;)printf 2^i++"+1/"}

Try it online!

Could remove the BEGIN and save 5 bytes if input were allowed. Using exponents definitely cheaper byte-wise than multiplication. :)

Hopefully 2^1023 is close enough to infinity (on my work computer). Unfortunately the TIO link truncates earlier than that (around 921). But 17726622920963562283492833353875882150307419319860869157979152909707315649514250439943889552308992750523075148942386782770807567185698815677056677116184170553481231217950104164393978236130449019315710017470734562946173533283208371259654747728689409291887821024109648618981425152 does seem pretty close to infinity. :)

Answer 28

Haskell - 66 62 60 51 49 chars

import Data.List
main=print$intercalate" + 1/"$map(show.(2^))[0..]

This prints the string constructed by printing the string representations of the powers of two starting from 1, separated by the string " + 1/".

The code itself is 49 bytes, the import and newline push it up to 66

Edit: (62)

Shaved 4 bytes by cutting out the import and defining intercalate with a much shorter name

f(x:xs)s=x++s++xs`f`s
main=print$map(show.(2^))[0..]`f`" + 1/"

Edit 2: (60)

Shaved 2 more characters by realizing I didn't need to use the (x:xs) list convention:

f(x:y)s=x++s++y`f`s
main=print$map(show.(2^))[0..]`f`" + 1/"

Edit 3: (51)

Reimplemented the definition of f and the map as the body of a fold to save 9 more characters

main=print$foldr((++).(++" + 1/").show.(2^))""[0..]

Edit 4: (49)

As was pointed out by Laokoni, I can remove the spaces to cut down 2 more bytes off the total:

main=print$foldr((++).(++"+1/").show.(2^))""[0..]

Answer 29

Braingolf, 41 37 bytes

Saved 4 bytes because I realised I don't need the spaces, always read the spec thoroughly kids

1!_V"+/">!@R.[<!_>2*v<!@>R!_v!@R<1+>]

Try it online!

Can probably be golfed better, but it works.

Answer 30

Lua 5.1 + BC, 50 bytes

Now works with arbitrary long integers.

I don't usually golf in Lua, so any tips are welcome

i=1::a::print(i,'+1/')i=require"bc".mul(i,2)goto a

Answer 31

Fourier, 20 bytes

1~io(43a1o47ai*2~io)

Try it online!

I think this may only work on Try it Online due to differences in how Python and Javascript handle large numbers.

Explanation Psuedocode:

i = 1
Print i
While i != 0
    Print "+1/"
    i = i * 2
    Print i
End While

Answer 32

Ruby 46 bytes

i=1;loop{print (i<2?"#{i}":"1/#{i}")+"+";i*=2}

any improvement suggestions are welcome -))

Answer 33

F#, 39 bytes

let rec f x=printf"%A+1/"x;x*2I|>f
f 1I

Try it online!

Answer 34

Labyrinth, 25 bytes

)
:}!43.4
"     9
*{2.74.

Try it online!

Answer 35

GNU Make, 45 57 bytes

Using shell:

$(eval 1?=1)$(info $1+1/)$(call $0,$(shell expr $1 \* 2))

Using GMSL (a lot slower, calculates in unary), a complete makefile — 66 bytes:

include gmsl
P=$(info $1+/)$(call P,$(call double,$1))
$(call P,1)

Answer 36

Common Lisp, 42 bytes

(do((x 1(* x 2)))(())(format t"~a+"(/ x)))

Try it online (TIO will show you partial output)

Explanation

(do ... )          ;looping
((x 1(* x 2))      ;define x as 1 and then in new interation steps 
                   ;as result of 2 * x
(())               ;no ending condition
(format t"~a+"...) ;print argument and "+"
(/ x)              ;divide 1 by x - output is like this: 1, 1/2, 1/4, ...

Answer 37

PHP, 29 bytes

for($a=.5;$a*=2;)echo"$a+1/";

Answer 38

Ruby, 24 bytes

a=1;a/=2r while$><<a<<?+

Try it online!

Heavily based on G B's answer. But rationals are fun :V

Answer 39

MSX-BASIC, 36 bytes

1?"1+":X=2
2?"1/";X;"+":X=X*2:GOTO2

It works up to 70368744177664, then it starts displaying the values in exponential notation due to the precision limitation of the language, then it overflows at 1e62.

Answer 40

Taxi, 916 bytes

2 is waiting at Starchild Numerology.'1' is waiting at Writer's Depot.Go to Writer's Depot:w 1 r 3 l 2 l.Pickup a passenger going to Post Office.Go to Post Office:n 1 r 2 r 1 l.Go to Starchild Numerology:s 1 r 1 l 1 l 2 l.Pickup a passenger going to Cyclone.Go to Cyclone:e 1 l 2 r.[r]Pickup a passenger going to Trunkers.Pickup a passenger going to Multiplication Station.Go to Trunkers:s 1 l.Go to Multiplication Station:e 1 r 4 l.Pickup a passenger going to Cyclone.Go to Cyclone:s 1 r 2 l 2 r.Pickup a passenger going to Multiplication Station.Pickup a passenger going to The Babelfishery.' + 1/' is waiting at Writer's Depot.Go to Writer's Depot:s.Pickup a passenger going to Post Office.Go to The Babelfishery:n 1 r 2 r 1 r.Pickup a passenger going to Post Office.Go to Fueler Up:n.Go to Post Office:s 1 r 1 l.Go to Trunkers:s 1 r 1 l.Pickup a passenger going to Cyclone.Go to Cyclone:w 2 r.Switch to plan "r".

Try it online! (TIO will output the first 893 terms (up to ~3.3×10²⁶⁸) before cutting it off.)

Ungolfed:

2 is waiting at Starchild Numerology.
'1' is waiting at Writer's Depot.
Go to Writer's Depot: west 1st right 3rd left 2nd left.
Pickup a passenger going to Post Office.
Go to Post Office: north 1st right 2nd right 1st left.
Go to Starchild Numerology: south 1st right 1st left 1st left 2nd left.
Pickup a passenger going to Cyclone.
Go to Cyclone: east 1st left 2nd right.
[r]
Pickup a passenger going to Trunkers.
Pickup a passenger going to Multiplication Station.
Go to Trunkers: south 1st left.
Go to Multiplication Station: east 1st right 4th left.
Pickup a passenger going to Cyclone.
Go to Cyclone: south 1st right 2nd left 2nd right.
Pickup a passenger going to Multiplication Station.
Pickup a passenger going to The Babelfishery.
' + 1/' is waiting at Writer's Depot.
Go to Writer's Depot: south.
Pickup a passenger going to Post Office.
Go to The Babelfishery: north 1st right 2nd right 1st right.
Pickup a passenger going to Post Office.
Go to Fueler Up: north.
Go to Post Office: south 1st right 1st left.
Go to Trunkers: south 1st right 1st left.
Pickup a passenger going to Cyclone.
Go to Cyclone: west 2nd right.
Switch to plan "r".

Explanation:

Pickup the string `1` and print it to stdout to get started.
Pickup the number `2` and duplicate it so we have one to keep around and one to print.
Abuse the truncation function to leave a `2` for later.
(It's en route and has a short name so it saves bytes which is really important in Taxi.)
Go multiply the running total by 2.
(The first time, it's just a `2` so it returns itself. After that, it's 2 times the running total.)
Pickup the string ` + 1/`.
Convert the multiplication result from a number to a string.
Get gas.
Print the string ` + 1/` and the string multiplication result to stdout.
Go pickup that `2` from Trunkers, copy it again, and go back to `[r]`.

The output has trailing zeros on every number: 1 + 1/2.000000 + 1/4.000000 + ... but, per a comment by OP, that's OK so long as it's the same number of zeros every time.

Answer 41

Cubix, 26 bytes

oruUroorp"/1+"1>O2uw"wo;;q 

A fairly linear program. Push characters /1+ onto the stack, push 1 onto the stack. Start loop, output number, multiply it by 2, push it to the bottom, pop the multipliers, output the characters on the stack and loop.

Try it here

Maps to

      o r u
      U r o
      o r p
" / 1 + " 1 > O 2 u w "
w o ; ; q * . . . . . .
. . . . . . . . . . . .
      . . .
      . . .
      . . .

Answer 42

Clojure, 59 bytes

#((fn[n](print(if(= 1M n)1(str 1\/ n))\+)(recur(* n 2)))1M)

A function that prints the fraction, doubles the denominator, then recurses.

(defn zenos-dichotomy-paradox []
  ; Create an anonymous function...
 ((fn [n]
    ; ... that prints the fraction (or just 1 if (= n 1))...
    (print
      (if (= 1M n) 1 (str 1 \/ n))
      \+)
    ; ..., doubles the denominator, then recurses infinitely.
    ; Will not SO since I'm using recur.
    (recur (* n 2)))
   ; Start the recursive function off with a 1. The M makes it a BigInteger so
   ;  I don't get an IntegerOverflow.
  1M))

Outputs

1 +1/2 +1/4 +1/8 +1/16 +1/32 +1/64 +1/128 +1/256 +1/512...

I can make the spacing more consistent at the cost of 2 bytes if necessary.

I tried to be clever and abuse Clojure's built in Ratio number type, but I got an overflow pretty quickly. 1/1 would have been automatically simplified to 1, getting rid of 1 as a special case. It would have been nice.

Answer 43

Mathematica, 45 bytes

Row@Flatten@Table[{1/2^t,"+"},{t,0,Infinity}]

Answer 44

cQuents, 13 bytes

|+1/=1:2^($-1

Try it online!

Explanation

|+1/            Set sequence join to "+1/"
    =1          Set first item in sequence to 1
      :         Mode : (sequence)
       2^($-1   Each item in the sequence equals 2 to the power of the index (1-based) of the current item minus 1
             )  Implicit closing parenthesis