Sums and Approximations This explains the Rule of 72: if you invest a dollar for n years at 72 /n perd cent interest, then the value of your dollar approaches e 2/100=2.05443 dollars from below as n goes to infinity. So the Rule overestimates your return for small n, slightly underestimates for large n, and is fairly accurate for typical values of n. A"Rule for 71"might be even more ccurate, but 72 has lots of convenient divisors, while 71 is inconveniently prime 3.3 Upper and Lower bounds Sometime an upper or lower bound on a function is needed instead of a simple approx- imation. For example, if you re computing the probability that a critical system fails,an upper bound would give you a reassuring, conservative answer. We can get good upper and lower bounds on many functions by looking at the error term in Taylor's Theorem Taylor's Theorem with one ordinary term and the error term says f(r)= f(0)+rf(a) for some z E0, r This is actually an equation, not an approximation. The problem is that we don' t know the value of z in the error term however we do have some bounds on 2, and those bounds imply upper and lower bounds on the function f(a) For example, suppose f(a)=In(1+r). Then Taylor's Theorem says ln(1+x)=0+x for some z E[0, z. If we assume r is positive, then the expression on the right is maximized when 2=0 and minimized when z=x. Therefore, we have the bounds ≤ln(1+x) The upper bound is the same as the linear Taylor approximation. Let's add the lowe bound to our plot: 1.5 y (1+x) x2/2 This lower bound is fairly accurate for small a. However, for large the lower bound /(1+a) asymptotically approaches 1 while the actual function In(1+.)diverges te InfinitySums and Approximations 13 This explains the Rule of 72: if you invest a dollar for n years at 72/n percent interest, then the value of your dollar approaches e72/100 = 2.05443 dollars from below as n goes to infinity. So the Rule overestimates your return for small n, slightly underestimates for large n, and is fairly accurate for typical values of n. A “Rule for 71” might be even more accurate, but 72 has lots of convenient divisors, while 71 is inconveniently prime. 3.3 Upper and Lower Bounds Sometime an upper or lower bound on a function is needed instead of a simple approx￾imation. For example, if you’re computing the probability that a critical system fails, an upper bound would give you a reassuring, conservative answer. We can get good upper and lower bounds on many functions by looking at the error term in Taylor’s Theorem. Taylor’s Theorem with one ordinary term and the error term says: f(x) = f(0) + xf (z) for some z ∈ [0, x] This is actually an equation, not an approximation. The problem is that we don’t know the value of z in the error term. However, we do have some bounds on z, and those bounds imply upper and lower bounds on the function f(x). For example, suppose f(x) = ln(1 + x). Then Taylor’s Theorem says: 1 x ln(1 + x) = 0 + x · 1 + z = 1 + z for some z ∈ [0, x]. If we assume x is positive, then the expression on the right is maximized when z = 0 and minimized when z = x. Therefore, we have the bounds: x x ≤ ln(1 + x) ≤ x 1 + The upper bound is the same as the linear Taylor approximation. Let’s add the lower bound to our plot: 6 1.5 y = x y = ln(1 + x) 1 y = x/(1 + x) 0.5 y = x − x2/2 0 - 0 1 2 3 This lower bound is fairly accurate for small x. However, for large x, the lower bound x/(1 + x) asymptotically approaches 1 while the actual function ln(1 + x) diverges to infinity