18.01 Review session, December 11, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
.
Final exam review, part #4: Estimates and comparisons
Taylor series approximation
• Taylor’s Theorem (with approximation):
f (x) = f (a) + f 0 (a)(x − a) +
1 00
1
1
f (a)(x − a)2 + · · · + f (n) (a)(x − a)n +
f (n+1) (ξ)(x − a)n+1
2!
n!
(n + 1)!
for some ξ between x and a. (The boxed term is the error term.)
◦ Useful for getting error bounds on the linear (or quadratic, or cubic, or. . . ) approximation of a
function.
Problems to review: pset 7 #3 and 4
Series and integral comparison
P
• If the an ’s do not go to 0, then
an does not converge. However, if you show that an → 0 this doesn’t
tell you anything about convergence!!!!!
P
• Series comparison:
|an | converges, then come up with |bn |’s such that
P if you want to show that
|bn | ≥ |an | and
|bn | converges.
P
◦ Divergence version:
|an | diverges, then come up with |bn |’s such that
P if you want to show that
|bn | ≤ |an | and
|bn | diverges.
• SomePseries you should know:
∞ 1
◦
n=1 n diverges
P∞ n
◦
n=1 r converges if |r| < 1, and diverges otherwise.
P∞ 1
◦
n=1 na converges if a > 1
R
• Integral comparison: if you want to show
R that |f | converges, then come up with g such that |g| ≥ |f |
on the range that you care about, and |g| converges.
R
◦ Divergence version: if you want to showRthat |f | diverges, then come up with g such that |g| ≤ |f |
on the range that you care about, and |g| diverges.
• Integrals
R ∞ you should know:
◦ 1 x1a dx converges for a > 1, and diverges otherwise.
R1
◦ 0 x1a dx converges for a < 1, and diverges otherwise.
R∞
P∞
•
1 f (n) converges ⇐⇒ 1 f (t) dt converges.
Problems to review: pset 9 #10, pset 11 #1-5
Problems
1. Consider approximating tan−1 (0.1) using a linear approximation at a = 0.
(a) Is this approximation greater than or less than the true value?
1
(b) Come up with error bounds for the approximation.
√
√
2. Your calculator says that e = 1.7082119 . . . , and your friend’s calculator says that e = 1.6487212 . . . .
Who has the faulty calculator?
Hint: use a quadratic approximation.
3. Do the following integrals converge?
Z ∞
1
(a)
dx
ln
x
2
Z
(b)
0
1
1
dx
x ex
4. Do the following series converge?
(a)
P∞
(e)
P∞
(b)
P∞
(f)
P∞
(c)
P∞
(g)
P∞
(d)
P∞
1
n=1 en
1
n=1 nen
1
n=2 n(ln n)2
n
n=2 n2 −1
n
n=1 en
1
n=1 (ln n)2
n
n=2 ln n
1a. >; b. (0.1 − (0.1)3 , 0.1); 2. you’re wrong; 3a. diverge; b. diverge; 4a. converge; b. converge; c. converge; d. diverge;
e. diverge; f. converge; g. diverge
2