18.01 Section, November 10, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
.
More series convergence, and introduction to integration
1. Find the radius of convergence of arctan x at a = 0.
2. Determine whether the following series converge:
∞
∞
X
X
n
sin2 n
(b)
(−1)n
(a)
2
n
n+1
n=1
3. Evaluate
R 2π
0
(c)
n=1
∞
X
(−1)n
n=1
en
n!
sin x dx. Verify your answer by re-interpreting the integral in terms of areas.
4. Find the following definite and indefinite integrals:
Z 1
Z
x
x
(a)
e dx
(b)
sin 5x + cos dx
2
−1
1
Z
(c)
3x2
dx
1 + x3
5. Bonus question: Show that
X cos( 2π n)
3
n
converges.
Review
•
Rb
f (x) dx is the (signed) area under the graph y = f (x).
Rb
• (Fundamental theorem of calculus) a f 0 (x) dx = f (b) − f (a)
P
P
•
ai converges absolutely if
|ai | converges.
P
P
• Dominated absolute P
convergence (series comparison):
bi looks worse than
ai in terms
P “if
of convergence, and
bi converges, then so does
ai .” More precisely, if:
◦ all the ai ’s and bi ’s are positive;
a
◦ ai ≤ bi for all i (or, all i ≥ N for some large N );
P
◦
bi converges;
then
P
ai converges.
P
• Alternating series test: i ai = 0 converges if:
◦ the ai alternate in sign;
◦ |ai | ≥ |ai+1 | for all i (good enough to have this for all i ≥ N for some large N );
◦ limi→∞ ai = 0.
P n
• Geometric series:
a converges iff |a| < 1, and in that case converges to
1
1−a .
• Ratio test (shows absolute
P convergence):
◦ Every power series
ai xi has a radius of convergence R;
ai exists, then it = R.
◦ if lim i→∞ ai+1 ◦ Taylor expansions:
x2 x4 x6
+
−
+ ...
2!
4!
6!
x3 x5 x7
arctan x ≈ x −
+
−
+ ...
3
5
7
1
1
1
ln x ≈ (x − 1) − (x − 1)2 + (x − 1)3 − (x − 1)4 + . . .
2
3
4
cos x ≈ 1 −
2
at a = 0
at a = 0
at a = 1