Homework

Problem 1.

Let {fn} be the sequence of functions on [0, 1] defined by fn(x) = nx(1−x4)n.

Show that {fn} converges pointwise. Find its pointwise limit.

Problem 2.

Is the sequence of functions on [0, 1) defined by fn(x) = (1 − x)

1

n pointwise

convergent? Justify your answer.

Problem 3.

Consider the sequence {fn} of functions defined by

fn(x) =

n + cos(nx)

2n + 1

for all x in R.

Show that {fn} is pointwise convergent. Find its pointwise limit.

Problem 4.

Consider the sequence {fn} of functions defined on [0, ] by fn(x) = sinn(x).

Show that {fn} converges pointwise. Find its pointwise limit. Using the

above theorem, show that {fn} is not uniformly convergent.

5

Problem 1.

Let {fn} be the sequence of functions on [0, 1] defined by fn(x) = nx(1−x4)n.

Show that {fn} converges pointwise. Find its pointwise limit.

Problem 2.

Is the sequence of functions on [0, 1) defined by fn(x) = (1 − x)

1

n pointwise

convergent? Justify your answer.

Problem 3.

Consider the sequence {fn} of functions defined by

fn(x) =

n + cos(nx)

2n + 1

for all x in R.

Show that {fn} is pointwise convergent. Find its pointwise limit.

Problem 4.

Consider the sequence {fn} of functions defined on [0, ] by fn(x) = sinn(x).

Show that {fn} converges pointwise. Find its pointwise limit. Using the

above theorem, show that {fn} is not uniformly convergent.

5

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