Sure, but if you see 1−2+3−4+5...=1/4, you already have all the intuition you need. The rest is detail.
The example I used, 1+2+4+8+…=−1, is the same kind of thing, a power series applied outside its domain of convergence, which I used instead because, while it doesn’t lend itself to the derivation directly, it looks more like the sum we seek (in particular, all positive integers on the left and a negative number on the right), and I expected the formula for an infinite geometric series to be more familiar to most readers.
Sure, but if you see 1−2+3−4+5...=1/4, you already have all the intuition you need. The rest is detail.
The example I used, 1+2+4+8+…=−1, is the same kind of thing, a power series applied outside its domain of convergence, which I used instead because, while it doesn’t lend itself to the derivation directly, it looks more like the sum we seek (in particular, all positive integers on the left and a negative number on the right), and I expected the formula for an infinite geometric series to be more familiar to most readers.