Why are sums like ∑ (x_n - x_(n +1)) called telescopic?
People usually say sums like that are telescopic or that they telescope. Well, the terms cancel out, so that we get x_1 - x_2 + x_2 - x_3 ....+ x_n - x_(n +1) = x_1 - x_(n + 1). It's beautiful, if x_n → 0 then the sum converges to x_1, kinda nice.
But, what does this have to do with a telescope? As far as I know, a telescope is an optical instrument, usually in the shape of a tube, containing lenses, and used to observe stars, planets, etc.The sky, in general. What on earth does a telescope have to do with series?
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Do you have a camera with focusing lens that works for both close and far photos ? ...what happens to the lens???...old time handheld telescopes compacted by pushing on both ends.....many umbrellas telescope to a smaller size..or from small to large
Just to add a bit of information to the previous answers: An illustration of a collapsing telescope can be found at the auction site below.
This telescope collapses down from a maximum of 14 inches to 5 inches in length.
it's mainly a clever name. a telescoping series- as you said, collapses to something simpler, as does a telescope.