Converging together, not one by one

Mathematics · Calculus & Analysis · ages 20-21

Name ______________________ Date ____________

Let f_n(x) equal x to the power n. What is f_10(0.5)?
Answer: ______________
Let f_n(x) equal x to the power n for x in the closed interval from 0 to 1. What is the pointwise limit function?
- 0 at every point of the interval
- 0 for x below 1 and 1 at x equal 1
- 1 at every point of the interval
- x at every point of the interval
Let f_n(x) equal x to the n on [0, 1]. What is the pointwise limit?
- 0 at every point
- 0 below 1 and 1 at 1
- 1 at every point
A student claims pointwise convergence means a single N works for every x at once. Is the student right?
Circle one: True False
Let g_n(x) equal x divided by n for x in [0, 10]. What is the largest value of |g_4(x) minus 0| on this interval?
Answer: ______________
A student sees a continuous pointwise limit and concludes the convergence is uniform. Is the student right?
Circle one: True False
Which sentence correctly describes uniform convergence of f_n to f on a set S?
- For every x in S and every positive epsilon there is an N such that n beyond N gives |f_n(x) minus f(x)| below epsilon
- There is an N such that for every positive epsilon, n beyond N gives |f_n(x) minus f(x)| below epsilon for every x
- For every positive epsilon there is an N such that n beyond N gives |f_n(x) minus f(x)| below epsilon for every x in S
- For every n there is a positive epsilon such that |f_n(x) minus f(x)| is below epsilon at every x
Let f_n(x) equal x to the power n on the half open interval [0, 1). Each f_n is continuous and the pointwise limit is the continuous zero function, so the convergence must be uniform. True or false?
Circle one: True False
A prover writes: given x, choose N bigger than 1 over (x times epsilon). Which convergence did they prove?
- Uniform convergence, since N was found
- Pointwise convergence, since the chosen N depends on x
- Divergence
A series of continuous functions converges pointwise on [a, b]. Which operation is licensed if the convergence is uniform, but not by pointwise convergence alone?
- Evaluating the sum at a single point
- Adding up the first ten terms
- Multiplying every term by a fixed constant
- Integrating the sum term by term over [a, b]

Answer key

For grown-ups. Fold this page away before handing over the rest.

Converging together, not one by one W1-mt_iAGk1ML5MA-s1

0.0009765625 · 0.5 to the power 10 equals 1 divided by 1024, which is 0.0009765625.
0 for x below 1 and 1 at x equal 1 · For x below 1, x to the power n tends to 0, while 1 to any power stays 1.
0 below 1 and 1 at 1 · Below 1 the powers die to 0, while at 1 they stay 1.
False · One shared N for all x is the uniform demand, not the pointwise one.
2.5 · g_4(x) equals x divided by 4, which is largest at x equal 10, giving 2.5.
False · A continuous limit never proves uniformity; one N for all x must still be checked.
For every positive epsilon there is an N such that n beyond N gives |f_n(x) minus f(x)| below epsilon for every x in S · Uniform convergence puts epsilon first, then one N that works for every x at once.
False · False. The largest gap between f_n and zero stays 1 for every n, so the convergence is not uniform.
Pointwise convergence, since the chosen N depends on x · An N built from x is pointwise by quantifier order.
Integrating the sum term by term over [a, b] · Uniform convergence lets the integral pass inside the infinite sum. The other operations need no convergence at all.