Problem 1: Determine the radius of convergence of the series }>_) a,x”, where 5”, if n is odd Qn=-41

Problem 1: Determine the radius of convergence of the series }>_) a,x”, where
5”, if n is odd
Qn=-41 . .
zr, if n is even.
Problem 2: Consider a power series )>~ )@n2” with radius of convergence R.
(a) Prove that if all the coefficients a, are integers and if infinitely many of them are
nonzero, then R < 1.
(b) Prove that if limsup,_,,,|@¢n| > 0, then R < 1.
Problem 3: Define functions f, : R > R by f,(z) = cos"(x). Note that these functions are
all continuous. Prove that
(a) limpsoo fn(x) = 0 if x is not a multiple of z.
(b) limps fa(z) = 1 if x is an even multiple of 7.
(c) limpsoo fr(z) does not exist if x is an odd multiple of z.
Problem 4: Let f,(2) = nz” for x € [0,1] and n € N. Show that
(a) limpsoo fn(x) = 0 for x € [0, 1).
(b) However, limnsoo f fr(a)dx = 1.
Problem 5: Let (f,) be a sequence of monotonically increasing functions which converges
pointwise to f. Show that f must also be monotonically increasing.

Problem 6: Define a sequence of functions f, : R — R by
l,ifn<a2r<n+l1
frlz) = .
0, otherwise.
(a) Find f(x) = limpsoo fn(2).
(b) Determine whether f,, > f uniformly on R. Justify your answer.
Problem 7: Define a sequence of functions f,, : R — R by
wen
fr(z) = T4227"
(a) Find f(x) = limp f(z).
(b) Determine whether f, — f uniformly on R. Justify your answer.
Problem 8: Let f, : R — R be a sequence of continuous functions, and (z,) be a sequence
in [a,b] converging to xo. If f, + f uniformly, then prove that f,(tn) > f (xo).

Problem 8: Let f,, : R — R be a sequence of continuous functions, and (z,,) be a sequence
in [a,b] converging to zo. If f, > f uniformly, then prove that f,(2n) > f (20).
Problem 9: Define a sequence of functions f;, : R > R by
1
x) = >~—s.
Sel) x? + k?
Prove that 5°”, f, converges uniformly on R.
Problem 10: Suppose that each g; is a continuous function on [a, b], and that the series 377", gx
converges uniformly to f on [a,b]. Prove that
b Oo Ab
/ f(a)dr = > / on (x)dr.
a k=1 a