Sums and Approximations In the second step, notice that the 1/(1+p) term in the numerator goes to zero in the limit. The third equation follows by simplifying Plugging in m=$40, 000 and p=0.035 into this formula gives V a $1, 182, 857. The value of money drops off so fast that even an infinite number of payments are worth on a bit over a million dollars. In fact, the total value of all payments beyond the 50th is only about$200,000! More generally, we can get a closed form for infinite geometric sums from Theorem 1 by taking a limit Corollary 2. If 2<1, then Pr lim lim 1 The first equation uses the definition of an infinite limit, and the second uses Theorem 1 In the limit, the term antl in the numerator vanishes since z<1 We now have closed forms for both finite and infinite geometric series. Some example are given below. In each case, the solution follows immediately from either Theorem 1 (for finite series)or Corollary 2(for infinite series) 248 ∑(1/2) (1/ 11 1/2)2 2 2+4-8 1+2+4+8 22 1+3+9+27+…+3-=∑3 Here is a good rule of thumb: the sum of a geometric series is approximately equal to the term with greatest absolute value. In the first two examples, the largest term is equal to 1 and the sums are 2 and 2/3, which are both relatively close to 1. In the third example, the sum is about twice the largest term. In the final example, the largest term is 3"- and the sum is(3n-1)/2, which is 1.5 times greater� � Sums and Approximations 5 In the second step, notice that the 1/(1 + p)n term in the numerator goes to zero in the limit. The third equation follows by simplifying. Plugging in m = $40, 000 and p = 0.035 into this formula gives V ≈ $1, 182, 857. The value of money drops off so fast that even an infinite number of payments are worth on a bit over a million dollars. In fact, the total value of all payments beyond the 50th is only about $200,000! More generally, we can get a closed form for infinite geometric sums from Theorem 1 by taking a limit. Corollary 2. If |z| < 1, then: � 1 ∞ i z = 1 − z i=0 Proof. ∞ n−1 i z = lim z i i=0 n→∞ i=0 n 1 − z = lim n→∞ 1 − z 1 = 1 − z The first equation uses the definition of an infinite limit, and the second uses Theorem 1. In the limit, the term zn+1 in the numerator vanishes since |z| < 1. We now have closed forms for both finite and infinite geometric series. Some examples are given below. In each case, the solution follows immediately from either Theorem 1 (for finite series) or Corollary 2 (for infinite series). ∞ 1 1 1 � 1 1 + + + + . . . = (1/2)i = = 2 2 4 8 1 − (1/2) i=0 ∞ 1 1 1 � 1 2 1 − + . . . = (−1/2)i = 1 − (−1/2) = 2 3 + 4 − 8 i=0 n−1 1 + 2 + 4 + 8 + . . . + 2n−1 = �2i = 1 − 2n = 2n − 1 1 − 2 i=0 n−1 3n 1 + 3 + 9 + 27 + . . . + 3n−1 = �3i = 1 − 3n = − 1 1 − 3 2 i=0 Here is a good rule of thumb: the sum of a geometric series is approximately equal to the term with greatest absolute value. In the first two examples, the largest term is equal to 1 and the sums are 2 and 2/3, which are both relatively close to 1. In the third example, the sum is about twice the largest term. In the final example, the largest term is 3n−1 and the sum is (3n − 1)/2, which is 1.5 times greater