Question 3 Suppose that f(x) nx2 1 + nx?' for all x ER. What is fo)...
bonse. Question 5 Suppose that f(x) nx2 1 + nx?' for all XER. What is lim f(x) for x =0?
7. Show that sin. nx-zs -zsin sin’x+ 1 4 3 sin' x sinº x +...= 2 sin x 2+sin x for all x ER
if f(t)={0
0<x<3
e^t x>3
QUESTION 4 r> 3 then fO S s s-1 35 s(s-1) d. (s-D
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 =...
Let fn (x) = 1 + (nx)? {n} are differentiable functions. (a) Show that {fn} converges uniformly to 0. (b) Show that .., XER, NEN. converges pointwise to a function discontinuous at the origin.
It is known that the Fourier series of f(x)=x is 2,6 21– 1)* * 'sin(nx) on [-1,1). n 1 1 1 Use this to find the value of the infinite series 1 - + + .... 3 5 7
It is known that Fourier series of f(x)=x is 2° 2(-1)" + "sin(nx) (n 1 on interval [-T, T). Use this to find the value of the infinite sum 1 - + 1 1 5 7 3
Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f be its summation function n sin(nx) b) Show that f E C(R) and that 1 cos(nx) f'(x)= 2-1 c) Show that 「 f#072821) f(x)dx = k=0
Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f...
1. (a) Let {fn}neN : [0,00) + R be a sequence of function define by: sin(nx) fn(x) 1+ nx (i) Guess the pointwise limit f of fn on (0,00) and justify your claim. [15 Marks] (ii) Show that fn + f uniformly on ſa, 00), Va > 0. [10 Marks) (iii) Show that fn does not converge uniformly to f on (0,00) [10 Marks] (Hint: Show that ||fr|| 21+(1/2) (b) Prove that a continuous function f defined on a closed...
Suppose that nx) 2x3-3x-X+ 2. Using only the theorem on bounds, find the smallest positive integer that is an upper bound for the zeros of f(x) and find the largest negative integer that is a lower bound for the zeros of fx). Upper bound 2, lower bound -2 Upper bound 1, lower bound -2 Upper bound 2, Lower bound 1 Upper bound 3, Lower bound -2 Upper bound = 3, Lower bound #-2 Question 8 (10 points) Find all the...