dmm,
@dmm@mathstodon.xyz avatar

Here's an interesting series:[S=\sum\limits_{n=1}^{\infty} {\left (\frac{a}{b}\right)}^{n}
]Does it converge, and if so, to what?

A few of my notes on all of this are here:
https://davidmeyer.github.io/qc/infinite_sum_a_over_b.pdf, and as always, questions/comments/corrections/* greatly appreciated.

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@dmm - why write a/b instead of just x? The answer depends only on a single number x = a/b, not on the separate numbers a and b.

dmm,
@dmm@mathstodon.xyz avatar

@johncarlosbaez It's true, but the problem (back when I first saw it) was posed in terms of (\frac{a}{b}) so I worked that way.

But you are right, I should do [S=\sum\limits_{n=1}^{\infty} x^n] (which converges to (\frac{x}{1-x}) for (|x| < 1), similar logic...)

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@dmm - Yes, I think it's always good to simplify things down to their essence.

dmm, (edited )
@dmm@mathstodon.xyz avatar

@johncarlosbaez I added the stuff in the figure. Thanks for the insight and help! -dmm

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