Let a n = ( 1 + 1 n ) n . (a) Show that if 0 ≤ a < b , then b n + 1 − a n + 1 b − a < ( n + 1 ) b n (b) Deduce that b n [ ( n + 1 ) a − n b ] < a n + 1 . (c) Use a = 1 + 1 / ( n + 1 ) and b = 1 + 1 / n in part (b) to show that { a n } is increasing. (d) Use a = 1 and b = 1 + 1 / ( 2 n ) in part (b) to show that a 2 n < 4 . (e) Use parts (c) and (d) to show that a n < 4 for all n . (f) Use Theorem 12 to show that lim n → ∞ ( 1 + 1 / n ) n exists. (The limit is e . See Equation 6.4.9 or 6.4 * .9 .
Let a n = ( 1 + 1 n ) n . (a) Show that if 0 ≤ a < b , then b n + 1 − a n + 1 b − a < ( n + 1 ) b n (b) Deduce that b n [ ( n + 1 ) a − n b ] < a n + 1 . (c) Use a = 1 + 1 / ( n + 1 ) and b = 1 + 1 / n in part (b) to show that { a n } is increasing. (d) Use a = 1 and b = 1 + 1 / ( 2 n ) in part (b) to show that a 2 n < 4 . (e) Use parts (c) and (d) to show that a n < 4 for all n . (f) Use Theorem 12 to show that lim n → ∞ ( 1 + 1 / n ) n exists. (The limit is e . See Equation 6.4.9 or 6.4 * .9 .
Solution Summary: The author explains how to use the binomial expansion to show bn+1-a
Consider the following system of equations, Ax=b :
x+2y+3z - w = 2
2x4z2w = 3
-x+6y+17z7w = 0
-9x-2y+13z7w = -14
a. Find the solution to the system. Write it as a parametric equation. You can use a
computer to do the row reduction.
b. What is a geometric description of the solution? Explain how you know.
c. Write the solution in vector form?
d. What is the solution to the homogeneous system, Ax=0?
2. Find a matrix A with the following qualities
a. A is 3 x 3.
b. The matrix A is not lower triangular and is not upper triangular.
c. At least one value in each row is not a 1, 2,-1, -2, or 0
d. A is invertible.
Find the exact area inside r=2sin(2\theta ) and outside r=\sqrt(3)
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