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a. Show that
limh→0 ((2 + h)/(2 - h))1/h = e.
b. Compute approximations to e using the formula N(h) = ((2+h)/(2−h))1/h, for h = 0.04, 0.02, and 0.01.
c. Assume that e = N(h)+K1h+K2h2 +K3h3 +· · · . Use extrapolation, with at least 16 digits of precision, to compute an O (h3) approximation to e with h = 0.04. Do you think the assumption is correct?
d. Show that N (−h) = N (h).
e. Use part (d) to show that K1 = K3 = K5 = · · · = 0 in the formula
e = N(h) + K1h + K2h2 + K3h3K4h4 + K5h5 +· · · ,
so that the formula reduces to
e = N(h) + K2h2 + K4h4 + K6h6 +· · · .
f. Use the results of part (e) and extrapolation to compute an O (h6) approximation to e with h = 0.04. |
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