Find $$1 + 2 + 4 + 8 + \cdots$$
Now I let $$s = 1 + 2 + 4 + 8 + \cdots$$ So, $$s = 1 + 2(1 + 2 + 4 + \cdots)$$ $$s = 1 + 2s$$ and this gives us: $$s = -1$$
Is there a more rigorous proof for this bizarre result or is this completely incorrect?
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3Before calculating the sum of an infinte series, you have to SHOW that the series converge. – Devansh Kamra Sep 23 '20 at 06:41
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Right. I get it now. – Siddhanth Iyengar Sep 23 '20 at 06:42
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1Does this answer your question? $1 + 2 + 4 + 8 + 16 \ldots = -1$ paradox – coudy Mar 05 '23 at 19:55
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You've shown that if the series converges to a finite value, that value is $-1$. Whether it converges depends on the norm applied to the error terms $\sum_{j=0}^{n-1}2^j-(-1)=2^n$. If we use the Euclidean norm, the trivial norm, or the $p$-adic norm with $p\ne2$, the series diverges; if we use the $2$-adic norm, the series converges to $-1$ as claimed.
J.G.
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