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Question 1: Let the functions M(n) and f(n) be defined as follows. if n = 0...
9. La ste) defined as follows 9. Let f(x) defined as follows: f(x) = 0 if x < -1 = 2(x + 1)/27 if - 1<x<2 = 2/9 if 2 < x < 5 = 0 otherwise. Find F(u) = f(x)dx, where u E R.
n=2
Question 3 3 pts Find the Fourier Sine series for the function defined by 0<c<n f() = { 0, 2n, n<3 < 2n and write down, 1. The period T and the frequency wo of the Fourier Sine series 2. The coefficients bn for n = 1,2,3,...
3. Let {Sn, n > 0} be a symmetric Random Walk on Z. Defined To inf(n > 1 : Sn-0} the time of first passage to state 0, prove that 2n - 1 for any n 2 1
0. For n E N, n > 1, let s, be defined by 8. *Let s1 1 +S2m 2 S2m-1 S2m+1 $2m 2 Find lim s, and lim s,.
5. Let f be defined on [0,1] by the following formula: 1 t = 1/n (n + N) 2n 0, otherwise (a) Prove that f has an infinite number of discontinuities in (0, 1). (b) Prove that f is nonetheless integrable on (0,1). (Hint: remember your geometric series!
2. (8 points) Let {fn}n>ı be a sequence of functions that are defined on R by fn(x):= e-nx. Does {{n}n>1 converge uniformly on [0, 1]? Does it converge uniformly on (a, 1) with 0 <a<1? Does it converge uniformly on (0, 1)?
5. Let be the function defined by f(x) = -1 3 1.5 if r <0 if 0<x<2 if 3 < r <5 Find the Lebesgue integral of f over (-10,10).
34.3 Let f be defined as follows: f(t) = 0 for t < 0; f(t) = t for 0 <t < 1; f(t) = 4 for t > 1. (a) Determine the function F(x) = $* f(t) dt. (b) Sketch F. Where is F continuous? (c) Where is F differentiable? Calculate F' at the points of differentiability.
Recall from class that the Fibonacci numbers are defined as follows: fo = 0,fi-1 and for all n fn-n-1+fn-2- 2, (a) Let nEN,n 24. Prove that when we divide In by f-1, the quotient is 1 and the remainder is fn-2 (b) Prove by induction/recursion that the Euclidean Algorithm takes n-2 iterations to calculate gcd(fn,fn-1) for n 2 3. Check your answer for Question 1 against this.
Recall from class that the Fibonacci numbers are defined as follows: fo =...
n=7
Question 3 3 pts Find the Fourier Sine series for the function defined by f(x) = { 0, 2n, 0 <*n n<<2n and write down, 1. The period T and the frequency wo of the Fourier Sine series 2. The coefficients for r = 1,2,3,...