The Sum of the Natural Numbers
Number Theory
Level
pending
Consider the infinite sequence of natural numbers, S = 1+2+3+4+5... What is the sum of the terms in this sequence?
(Give the answer to three significant figures due to the way Brilliant accepts answers)
The answer is -0.0833.
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1 solution
To solve this puzzle, we first need to consider a separate sequence entirely, lets call it X: X = 1-1+1-1+1.... We can find the sum of this sequence by simply adding it to itself and shifting it along one place (We can do this as it is an infinite series) and hence we get: 2X = 1+0+0+0+0... Therefore: X=0.5
Then, we take another infinite sequence, lets call it Y: Y = 1-2+3-4+5 ... Then, again we add it to itself, shifted along one place: 2Y = 1-1+1-1+1... As you can see, this is the same as our series X, therefore: 2Y = X And so Y=0.25
Next, we take our original series S, and subtract from it Y: S-Y = 0+4+0+8+0...
As you can see, this sequence is the sum of the natural numbers, multiplied by 4, therefore: S-Y = 4S Which we can rearrange to: -Y = 3S Then we substitute in the value for Y we found earlier to get: -0.25 = 3S Therefore S = -0.25/3 S = -1/12 S = -0.0833 (3sf)