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  "body": "\n## Summary\n\nThe rate of convergence of a convergent sequence measures how fast its distance to the limit shrinks.\nOne compares the error at step $k+1$ with a power of the error at step $k$.\nThe exponent is the order of convergence, such as linear or quadratic.\nIt is the standard way to compare iterative methods in numerical analysis.\nTwo examples are [[fixed-point iteration|K6ZF91051B]] and [[Newton's method|07EXVNF9VP]].\n\n## Explanations\n\n### L1 human\n\nTwo sequences can have the same limit and still approach it at very different speeds.\nWith linear convergence, each step multiplies the error by about the same factor, so one gains correct digits at a steady pace.\nWith quadratic convergence, each error is about the square of the previous error, so the number of correct digits roughly doubles at each step.\nFor example, $2^{-k}$ tends to $0$ linearly, while $2^{-2^k}$ tends to $0$ quadratically.\n",
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