18.01 Section, November 11, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
.
Infinite series and convergence criteria
Special instructions: Talk to your neighbor(s) about which of the three tests/criteria can be
used to solve problems 1 and 2.
1. Find the radius of convergence of the following Taylor series, and give the actual interval
on which it converges.
Relevant Taylor expansions can be found at the end of this worksheet.
(a) cos x (at a = 0)
(b) ln x (at a = 1)
(c) ln x (for arbitrary a)
Helpful fact: if f (x) = ln x, then f (n) (x) =
(−1)n−1 −n
x .
n−1
2. Do the following series converge? (why/ why not?)
(a)
X sin(4n)
n
4n
1
(b)
X sin x + cos x
n!
P
P
3. If
cn xn has a radius of convergence
2 and
dn xn has radius of convergence 3, what is
P
n
the radius of convergence of (cn + dn )x ?
4. Bonus problem: come up with a variant of the ratio test to deal with series
only even terms).
2
P
an x2n (i.e.
Review
•
P
ai converges absolutely if
P
|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;
◦ ai ≤ bi for all i (or, all i ≥ N for some large N );
P
◦
bi converges;
P
then
ai converges.
P
• Alternating series test: i ai 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!
1
1
1
ln x ≈ (x − 1) − (x − 1)2 + (x − 1)3 − (x − 1)4 + . . .
2
3
4
cos x ≈ 1 −
3
at a = 0
at a = 1