Let a1 = a,a2 = f(a1),a3=f(a2)=f(f(a)),,an+1=f(an),where a is some number and f
ID: 2869349 • Letter: L
Question
Let a1 = a,a2 = f(a1),a3=f(a2)=f(f(a)),,an+1=f(an),where a is some number and f is a continuous function. If = L, show that f(L) = L. Now, let a = 1. Find an example of a function f such that the corresponding sequence converges and such that f(x) = x has one solution. Find an example of a function f such that the corresponding sequence converges and such that f(x) = x has more than one solution. Find an example of a function f such that the corresponding sequence diverges and such that f(x) = x has no solution. Find an example of a function f such that the corresponding sequence diverges and such that f(x) = x has a solution.Explanation / Answer
A)f(x) =
sin(nx + 3)/?n + 1
for all x in R
The sequence converge such that f(x)= x
B)fn(x) =
cos^n (x) for ??/2 ? x ? ?/2. it is uniform convergence of the sequence and has many solution.
c)Let {fn} be the sequence of functions on R defined by fn(x) =nx. This sequence does not converge pointwise on R because lim
n??
fn(x) = ?
for any x > 0
d)fn(x) =nx/1+n^2 x^2
Sequence diveges
But for any ? < 1/2
And has a f(x) = x a solution.
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