Exam 3 Solutions 1. Using the bilinearity of the inner product we have x+y
2
xy
2
= x + y, x + y x y, x y
= x, x + 2 x, y + y , y ( x, x + 2 x, y + y, y ) = x, x + 2 x, y + y , y ( x, x 2 x, y + y , y ) = 4 x, y .
2. For (x, y ) = (0, 0) we have
2 2 f =
Exam 2 Solutions
1. False. For example, the odd terms of the alternating harmonic series
n=1
(1)n1 form the n
series
k=1
1 which diverges by the Integral Test. 2k 1
23 , 24 119 . 120 (n+1)!1 (n+1)!
5 2. (i) s1 = 1 , s2 = 6 , s3 = 2
and s4 = =1
1 2
It see
Exam 1 Solutions 1. Claim(1). an an+1 for all n 1. Proof. By induction on n. Since a1 = 2 < 2 + 2 = a2 the base step is valid. Now assume that am am+1 for some m 1. Then am+1 = 2 + am 2 + am+1 = am+2 so the induction step is also valid. Proof. By inductio