Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

• In the discussion of the Michelson–Morley experiment, we twice used the binomial approximation

  (1.58)

which holds for any number n and any x much smaller than 1 (that is, |x| ≪ 1). (In the examples, n was −1 and and x = β2 was of order 10−8.) The binomial approximation is frequently useful in relativity, where one often encounters expressions of the form (1 − x)n with x small. Make a table showing (1 − x)n and its approximation 1 − nx for and x = 0.5, 0.1, 0.01, and 0.001. In each case find the percentage by which the approximation differs from the exact result.