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Explain why calculating this series could cause paradox?

Using the power series expansion $\ln(x+1)=\sum _{k=1}^\infty \frac{(-1)^{k+1}}{k}x^k$ we have $\ln(2)=(1-\frac{1}{2})+(\frac{1}{3}-\frac{1}{4})+\cdots+(\frac{1}{2k-1}-\frac{1}{2k})+\cdots=\sum _{k=1}^\infty (\frac{1}{2k-1}-\frac{1}{2k})$. Let's rearrange the terms of this sequence so that two negative terms follow each positive term: $\ln(2)=(1-\frac{1}{2}-\frac{1}{4})+(\frac{1}{3}-\frac{1}{6}-\frac{1}{8})+\cdots+(\frac{1}{2k-1}-\frac{1}{4k-2}-\frac{1}{4k})+\cdots=\sum _{k=1}^\infty (\frac{1}{2k-1}-\frac{1}{4k-2}-\frac{1}{4k})$. However, $\frac{1}{2k-1}-\frac{1}{4k-2}-\frac{1}{4k}=\frac{1}{2}(\frac{1}{2k-1}-\frac{1}{2k})$; therefore, $\sum _{k=1}^\infty (\frac{1}{2k-1}-\frac{1}{4k-2}-\frac{1}{4k})=\frac{1}{2}\sum _{k=1}^\infty (\frac{1}{2k-1}-\frac{1}{2k})$, but how can the latter inequality hold if $\sum _{k=1}^\infty (\frac{1}{2k-1}-\frac{1}{2k})$ and $\sum _{k=1}^\infty (\frac{1}{2k-1}-\frac{1}{4k-2}-\frac{1}{4k})$ are equal to $\ln(2)$?

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Community
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The Substitute
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3 Answers3

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Nice argument! You have produced a special case of the Riemann Rearrangement Theorem.

If we have a conditionally convergent series, the terms can be rearranged so that the sum is anything we like, or so that the sum does not exist. You have rearranged a conditionally convergent series for $\log 2$ so that it has sum $\frac{1}{2}\log 2$.

Remark: Results that are clear for finite sums do not necessarily extend to infinite "sums." And continuous functions do not necessarily behave like the smooth curves of our imagination. That is part of what makes analysis necessary.

André Nicolas
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The series for $\ln 2$ is a conditionally convergent series. An absolutely convergent series is one that still converges if you take the absolute value of all the terms. This one is not. If you take the absolute value of all the terms, you get the harmonic series, which diverges. Only absolutely convergent series can be rearranged in order and still sum to the same value. Rearranging conditionally convergent series can give any value you want, as shown by the Riemann rearrangement theorem.

Ross Millikan
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You have rediscovered the difference between absolute convergence and conditional convergence, and the fact that rearrangement of conditionally convergent series can alter the sum.

If the sum of the positive terms and the sum of the negative terms are both finite, then the series converges absolutely and rearrangement won't change the sum.

Michael Hardy
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