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## Homework Statement

With binomial expansions,

[itex]\frac{x}{1-x}[/itex] = [itex]\sum[/itex][itex]^{n=\infty}_{n=1}[/itex]x

^{n}

[itex]\frac{x}{x-1}[/itex] = [itex]\sum[/itex][itex]^{n=\infty}_{n=0}[/itex]x

^{-n}

Adding these series yields:

[itex]\sum[/itex][itex]^{n=\infty}_{n=-\infty}[/itex]x

^{n}=0

This is nonsense, but what went wrong here?

## The Attempt at a Solution

Obviously, [itex]\frac{x}{1-x}[/itex]+[itex]\frac{x}{x-1}[/itex]=0 (1)

It's clear that they tried to transform [itex]\frac{x}{x-1}[/itex] = [itex]\sum[/itex][itex]^{n=\infty}_{n=0}[/itex]x

^{-n}into [itex]\frac{x}{x-1}[/itex] = [itex]\sum[/itex][itex]^{n=0}_{n=-\infty}[/itex]x

^{n}and then substitute into equation (1) and get an answer of zero.

From there, I'm not sure how in the world they manipulated that series to get a neg. infinity to show up in the limits, and where the mistake in that is.