Sequences, Series and Power Series
Below you will find the exercises we suggest you work on in connection with the Sequences, Series and Power Series theme and the project problems which are to be done for the exercise classes during October November 18-22 and November 25-29.
Suggested Exercises
9.1 Sequences and Convergence
9.1.1, 9.1.3, 9.1.24, 9.1.31, 9.1.34, 9.1.36.
9.2 Infinite Series
9.2.1, 9.2.10, 9.2.11, 9.2.16, 9.2.19, 9.2.21.
9.3 Convergence Tests for Positive Series
9.3.1, 9.3.4, 9.3.7, 9.3.9, 9.3.18, 9.3.28, 9.3.38.
-:Convergence tests
-:Problem 9.3.28
9.4 Absolute and Conditional Convergence
9.4.1, 9.4.5, 9.4.13, 9.4.20.
9.5 Power Series
9.5.1, 9.5.5, 9.5.12, 9.5.13, 9.5.15, 9.5.21.
-:Power series
-:Problem 9.5.12
9.6 Taylor and Maclaurin Series
9.6.1, 9.6.2, 9.6.7, 9.6.29, 9.6.37.
9.7 Applications of Taylor and Maclaurin Series
9.7.1, 9.7.15, 9.7.21, 9.7.24.
Project Problems
Week 47
These problems are to be presented November 18-22. See here for where you should meet, and here to sign up for presenting a problem.
Problem 1: Maple TA
Consider the sum
\[\sum_{n=1}^{\infty} \frac{1}{2n-1}\left(\frac{ax+3}{-4}\right)^n,\] where \(a\) is a positive, real number.
Determine the values of \(x\) for which the series
i) converges absolutely,
ii) converges conditionally,
iii) diverges.
Hint: :Convergence tests
Problem 2: Exam 1998 in SIF 5003, problem 6
a) Show that the series \[\sum_{n=2}^{\infty} (-1)^{n+1}\left(\frac{n-1}{n^2}\right)\] converges. Is the convergence absolute or conditional?
b) The partial sum \[S_9 = \sum_{n=2}^{9} (-1)^{n+1}\left(\frac{n-1}{n^2}\right)\]
is an approximation to the sum \(S\) of the series in a). What can you say about \(|S_9-S|\)?
Problem 3: Induction
Use induction to show that \[\sum_{i=0}^{n} x^i =\frac{1-x^{n+1}}{1-x}\] for \(n\geq 1\) and \(x \neq 1\).
Hint: Pink box, page 110 in Adams.
Problem 4: Exam 2003 in TMA4100, problem 10
A figure is made in the following way: We start with a rectangle with sides \(x\) and \(y\), and then add a smaller rectangle with sides \(\frac{1}{2}x\) and \(\frac{1}{2}y\), and then add another with sides \(\frac{1}{4}x\) and \(\frac{1}{4}y\) and so forth, ad infinitum, as indicated in the figure below.
The circumference of the figure is 6. What values of \(x\) and \(y\) makes the area maximal?
Hint: :Power series
Week 48
These problems are to be presented November 25-29. See here for where you should meet, and here to sign up for presenting a problem.
Problem 1: Maple TA
Find the sum of the series
\[\sum_{n=2}^{\infty} \frac{3}{n(n-1)7^n}\]
HINT: First find the sum of
\[\sum_{n=2}^{\infty} \frac{3x^n}{n(n-1)}\] by differentiating the series twice, and then integrating again. See also :Problem 9.5.12.
2: Exam December 2005, problem 7
a) Find the Taylor series at \(x=0\) for the function
\[\frac{\cos{x}-1}{x^2}. \] For which \(x\) does the series converge? You may use that \[\cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}+\cdots +(-1)^{n}\frac{x^{2n}}{(2n)!} \cdots\]
b) Use the series you found to calculate the integral \[\int_{0}^{1}\frac{\cos{x}-1}{x^2} \, dx\] with an absolute error less than \(10^{-4}\). Hint: :Problem 9.3.28.
3: Exam December 2009, problem 3
For which \(x\) does the series \[\sum_{n=1}^{\infty}\frac{(x+1)^n}{n2^n}\] converge?
4: Exam August 2013, problem 5
Consider the series
\[\sum_{n=1}^{\infty}\frac{x^{n+1}}{n} \]
a) Determine the radius of convergence \(R\) for the series, and determine if the series converges for \(\pm R \).
Hint: :Convergence tests.
b) Let \[f(x) = \sum_{n=1}^{\infty}\frac{x^{n+1}}{n}, \] for \(-R<x<R\). Find a closed expression for \(f\). HINT: Write \(f(x)=xg(x)\), and find a closed expression for \(g'(x)\).
5: Exam December 2011, problem 7
a) Integrate the geometric series
\[\frac{1}{1-x^2}=\sum_{n=0}^{\infty} x^{2n}, \, \, |x|<1 \] to show that
\[\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)=\sum_{n=0}^{\infty}\frac{x^{2n+1}}{2n+1}.\]
b) Show that for \(0<x<1\) we have that
\[\left|\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)-\sum_{n=0}^{N}\frac{x^{2n+1}}{2n+1}\right|<\frac{x^{2N+3}}{2N+3}\left(\frac{1}{1-x^2}\right), \] and use this to calculate \(\ln{2} \) with an error less than \(10^{-5}\).
6: Exam December 2010, problem 7
Show that the series \[\sum_{n=0}^{\infty}\frac{x^{3n+2}}{(3n+2)n!} \] converges for all \(x\) and that the sum is \[\int_{0}^{x}t\mathrm{e}^{t^3} \, dt. \]
7: Exam August 2010, problem 6
For what values of \(x\) does the series \[\sum_{n=0}^{\infty} \left(\frac{nx}{1+n}\right)^n \] converge?