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If a series such as '$a$' below adds to infinity:

$a = 1 + 2 + 4 + 8 + 16 + \cdots\to \infty$

Multiplying '$a$' by $2$ yields:

$2a = 2 + 4 + 8 + 16 + \cdots\to \infty$

However when I subtract these two series, I find a paradoxical answer. Series '$a$', which supposedly adds to infinity also equals -1.

$2a - a = (2 + 4 + 8 + 16 + \cdots) - (1 + 2 + 4 + 8 + 16 +\cdots)$

$2a - a = -1 + (2 - 2) + (4 - 4) + (8 - 8) + \cdots$

$a = -1 + 0 + 0 + 0 + 0 + \cdots$

$a = -1$

Does this mean that infinity equals $-1$?

Hanul Jeon
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Adil
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    Yes. See this question. – MJD Apr 06 '13 at 01:19
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    This has been asked and answered many times on this site: Infinity = -1 paradox, also http://math.stackexchange.com/questions/193872/prove-1-2-4-8-dots-1 , http://math.stackexchange.com/questions/88231/proof-that-1-infty , http://math.stackexchange.com/questions/90899/is-sum-limits-n-0-infty2n-1 – MJD Apr 06 '13 at 01:21
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    There are many arguments for assigning the formal object $1+2+2^2+2^3+\cdots$ the value $-1$. However, if we want to assign to $a_0+a_1+a_2+\cdots$ a value based on the limit of partial sums of the series, it cannot be done for $1+2+2^2+\cdots$. – André Nicolas Apr 06 '13 at 01:24

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