Let’s assume we are adding 2 + 2^2 + 2^3 +……..+2^n = now here , according to the sum of an infinite gp formula, which is given by : a/1-r, the answer to the summation will be negative two (-2) , just how is that possible ? we are not using any negative number, neither any subtraction and only natural numbers 1 through infinity, how do we get a negative number and that too just something as small as -2 . I know you can prove it mathematically but how do i really grasp it when all my natural instincts are telling me otherwise?
In: Mathematics
A core requirement of this formula is that |r| must be < 1. In this case, it is 2, and the “otherwise” clause of this formula states that the series _diverges_ to ±∞.
You are applying the formula in a case it does not. Take a look at the finite(!) sum formula
1 + r + r^^2 + r^^3 + … + r^^n = ( 1-r^^n+1 ) / ( 1-r ).
Notice the extra -r^^n+1 at the numerator. If n is huge, or we take the infinite limit, then the value of this extra term depends on r:
– if |r| < 1, then r^^n+1 gets very tiny and goes to 0.
– if |r| > 1, then r^^n+1 gets huge and goes to ±∞.
– if r = 1 then it stays 1; if r = -1 it alternates in sign.
So if you plug slightly abusively n = ∞ in, then you actually get
1 + r + r^^2 + r^^3 + … = 1/(1-r) – r^^∞ / (1-r)
and if now r = 2 then suddenly this becomes
1/(1-2) – 2^^∞ / (1-2) = -1 + 2^^∞ .
But the right hand side is obviously _huge_, infinite. Which is the same for the initial sum
1 + 2 + 4 + 8 + 16 + … = ∞.
So… actually no surprises here! The error in your formula is that you ignored that other half with the r^^n+1 in a case where it matters, and obviously leaving out the largest term hugely changes the result; in this case by ∞.
**tl;dr:** don’t use formulas that are only meant for special cases (here: |r| < 1) in other cases.
Side note for those interested: in other settings such as the _2-adic numbers_ we actually have |2| = 0.5 < 1 and the sum really converges to 1 + 2 + 4 + 8 + 16 + … = -1. This is not that weird if one also shows that there can not be a meaningful notion of “<” or “>” on those numbers, similar to how it is impossible on the complex ones.
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