Consider the function fx sigmak 1infinity sin xkk Where is

Consider the function f(x) = sigma_k = 1^infinity sin (x/k)/k Where is f defined? Continuous? Differentiable? Twice-differentiable?

Solution

the nth term of the function f(x ) is = [sin(x/k)]/k , where k = 1,2,3,4,5,6............infinity

now f is defined for all x E R . that is f is defined for all the x values which lie within the set of real numbers.

provided k = 1,2,3,4,5,6............infinity

likewise the function f has now breaking point on its graph so its continuous for all x E R , provided k = 1,2,3,4,5,6............infinity

f is differentiable for all x E R , provided k = 1,2,3,4,5,6............infinity

f is twice differentiable for all x E R , provided k = 1,2,3,4,5,6............infinity