advertisement

Math 166Z Homework # 7 Due Thursday March 24th 1. Consider the sequence an = n+1 n which converges to 1. (a) find a number N such that |an − 1| < .1 for all n ≥ N . (b) find a number N such that |an − 1| < .01 for all n ≥ N . (c) find a number N such that |an − 1| < .001 for all n ≥ N . 2. Does the sequence an = ln(n + 1) − ln(n) converge? If so, what number does an converge to? 1 + 2 + 3 + ··· + n converges to 12 . n2 Hint: Can you think of another way to write 1 + 2 + 3 + · · · + n? (you might want to look in section 4.1) 3. Show that the sequence an = 4. For the following, determine whether the series is convergent of divergent. If it is convergent, find its sum. ∞ X 5n (a) 8n n=0 (b) (c) ∞ X 4n+1 n=0 ∞ X n=1 (d) 5n 1 2n ∞ X 2n n=3 7n 5. For the following, find the values of x for which the series converges. What is the sum of the series for these values of x? ∞ X (a) (x − 3)n (b) n=0 ∞ X n=2 xn 5n 6. Write the following repeating decimals in the form of ab where a and b are integers. Hint: Re-write them as an infinite series and then find the sum of that series. (a) 0.1 = 0.11111 . . . (b) 0.123 = 0.1232323 . . . (c) 0.045 = 0.0454545 . . . (d) 0.19 = 0.199999 . . . 7. The present value (PV) of a payment in the future is given by the formula x PV = (1 + i)n where: x = the amount of the payment amount, i = the interest rate n = the number of time periods in the future the payment is made If I decide to give you $100 every year from now on (forever, these payments will eventually be made to your decedents) and the interest rate is 5% per year, what is the present value of these payments? PV = ∞ X n=1 100 (1 + .05)n (It may seem kind of strange, but this kind of payment was used in the past. It is called a perpetuity. The government of the United Kingdom used them in the 1750s to help reduce what it had to pay on its debt. It is still paying some of them today.) Page 2