Prepare for your exams
Get points
Guidelines and tips
Sell on Docsity
Docsity AI

Prepare for your exams

Study with the several resources on Docsity

Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips

Sell on Docsity

Docsity AI

Log in Sign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search for your university

Find the specific documents for your university's exams

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents

20 Points

For each uploaded document

Answer questions

5 Points

For each given answer (max 1 per day)

All the ways to get free points

Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

Week #8 Practice Quiz Solutions for Math 412: Convergence of Series and Power Series - Pro, Quizzes of Mathematics

California State University (CSU) - Fullerton Mathematics

Prof. Scott Annin

The solutions to problem 1 and problems 2(a) and 2(b) from the week #8 practice quiz of math 412. The problems involve determining the values of complex numbers z for which certain series converge, finding the radii of convergence for power series, and proving the relationship between the radii of convergence for related power series.

Typology: Quizzes

Pre 2010

Uploaded on 08/16/2009

koofers-user-cqa 🇺🇸

10 documents

1 / 2

This page cannot be seen from the preview

Don't miss anything!

March 16-20, 2009 Week #8 Practice Quiz Name:

Math 412

SOLUTIONS

Problem 1. (6 points) Determine all values of zsuch that

∞

X

n=0 z

1 + zn

converges, and for points of convergence, determine the sum.

SOLUTION: This is a geometric series, so convergence hinges upon the modulus of the common

ratio, z

1+z. More precisely, the given series converges if and only if



z

1 + z



<1⇐⇒ |z|<|1 + z|

⇐⇒ |z|2<|1 + z|2

⇐⇒ zz < (1 + z)(1 + z) = 1 + zz +z+z

⇐⇒ z+z > −1

⇐⇒ Re(z)>−1

2.

Thus, the given series converges if and only if Re(z)>−1

2. The value of the sum is

1

1−z

1+z

=z+ 1.

2

Problem 2. (4 p oints each) Determine the radius of convergence of each power series

below:

(a):

∞

X

n=0 n!

1000nzn

SOLUTION: Using the ratio test, we find that



cn+1

cn



=



(n+ 1)!1000n

n!(1000n+1)



=



n+ 1

1000 



→ ∞,

as n→ ∞. Thus, ρ= 0. 2

(b):

∞

X

n=0 n3(n+ 1)n

(3n)n(z−i+ 3)n.

Discover Quizzes of Mathematics California State University (CSU) - Fullerton

Partial preview of the text

Download Week #8 Practice Quiz Solutions for Math 412: Convergence of Series and Power Series - Pro and more Quizzes Mathematics in PDF only on Docsity!

March 16-20, 2009 Week #8 Practice Quiz Name:

Math 412

SOLUTIONS

Problem 1. (6 points) Determine all values of z such that

∞ ∑

n=

z

1 + z

)n

converges, and for points of convergence, determine the sum.

SOLUTION: This is a geometric series, so convergence hinges upon the modulus of the common

ratio,

z 1+z

. More precisely, the given series converges if and only if

z

1 + z

< 1 ⇐⇒ |z| < |1 + z|

⇐⇒ |z|

2 < |1 + z|

2

⇐⇒ zz < (1 + z)(1 + z) = 1 + zz + z + z

⇐⇒ z + z > − 1

⇐⇒ Re(z) > −

Thus, the given series converges if and only if Re(z) > −

1 2

. The value of the sum is

z 1+z

= z + 1.

Problem 2. (4 points each) Determine the radius of convergence of each power series

below:

(a): ∞ ∑

n=

n!

n

z

n

SOLUTION: Using the ratio test, we find that

cn+

cn

(n + 1)!

n

n!( n+ )

n + 1

as n → ∞. Thus, ρ = 0. 2

(b):

∞ ∑

n=

n 3 (n + 1) n

(3n) n

(z − i + 3)

n .

SOLUTION: To apply the root test, we must examine

|cn|

1 /n

n

3 /n (n + 1)

3 n

n

n 3 | ·

n + 1

3 n

as n −→ ∞, where the limit of

n

n 3 (which was 1) is determined by using logarithms and L’Hopital’s

Rule. Thus, L =

1 3

in the root test, and we find that ρ = 3. 2

Problem 3. (6 points) Assume that the radius of convergence of the power series

∞ ∑

n=

cnz

n

is ρ, where 0 < ρ < ∞. Prove that the radius of convergence of the power series

∞ ∑

n=

c

2 n z

n

is ρ

2 .

SOLUTION: According to the ratio test, we know that

ρ =

lim n→∞

cn+

cn

Now, by the ratio test, the power series ∞ ∑

n=

c

2 nz

n

converges if and only if

lim n→∞

c 2 n+ z n+

c 2 nz

n

= |z|

lim n→∞

c 2 n+

c 2 n

Hence, its radius of convergence is

lim n→∞

c

2 n+

c 2 n

lim n→∞

cn+

cn

) 2 =^ ρ

2 .

Week #8 Practice Quiz Solutions for Math 412: Convergence of Series and Power Series - Pro, Quizzes of Mathematics

Related documents

Partial preview of the text

Download Week #8 Practice Quiz Solutions for Math 412: Convergence of Series and Power Series - Pro and more Quizzes Mathematics in PDF only on Docsity!

March 16-20, 2009 Week #8 Practice Quiz Name:

Math 412

SOLUTIONS

1 /n