Universidad Carlos III de Madrid
Escuela Polit´ecnica Superior
Departamento de Matem´
aticas
First Course. Bachelor’s Degree in
Telematics, Communication System and Audiovisual System Engineering
Calculus I. Second Test, November 25, 2008 GRADE
Surname............................................................. Name..............................
D.N.I. ............................................................. Group ..............................
Time length: 80 min.
1. (Class problem) Approximate eusing a Taylor Polynomial, with error less than 0.001.
How many terms must we use in the polynomial?
[2 p.]
2. (Problem 3.1.4) Compute the limit
lim
n→∞
1 + 1
2+···+1
n
log n
[2 p.]
3. (Problems 3.2.1, 3.2.4) Analyze the convergence of the following series
1.
∞
X
n=2
log ³n+ 1
n´
2.
∞
X
n=1
(−1)n³arctan 1
n´2
Hint: arctan x=x−x3
3+x5
5−x7
7+· ·· + (−1)nx2n+1
(2n+ 1).[4 p.]
4. (Problem 3.3.4) Compute the Taylor Series, the radius and the interval of convergence of
f(x) = 1
2−x2
Hint: 1
1−x= 1 + x+x2+x3+···+xn··· .
[2 p.]
