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- $\infty = -1 $ paradox 7 answers
I was solving a recurrence problem which had a sequence such as $y = (1+2+4+8+...)\sqrt n$, and I wanted to find what $x = 1+2+4+8+...$ was. So consider $x = 1+2+4+8+...$ as an infinite series.
$$x-1 = 2+4+8+...$$
$$2(x)=2(1+2+4+8+...) = 2+4+8+... = x-1$$
$$2x = x-1 \Rightarrow x=-1$$
What does $x=-1$ represent? Have I made a mistake in my calculations? Of course, we expect that $x=\infty$ since the sum grows by a factor of $2$ with each term.