11.8: Power Series!
Wednesday, March 11
Speed Round
Determine whether each of the following series converges or diverges.
1.
∞
X
1
n
n=1
5.
∞
X
1
√
n
n=1
2.
∞
X
(−1)n
n
n=1
6.
∞
X
(−1)n
1 + ln n
n=1
10.
3.
∞
X
2n
n!
n=1
7.
∞
X
n2 ln n
4n
n=1
11.
3n
√
n2 + n
n=1
4.
∞
X
2n
n2
n=1
8.
∞
X
n5 5n
n!
n=1
12.
∞
X
n!
30n
n=1
9.
∞
X
(−2)n
n3
n=1
∞
X
(−1)n
n=1
∞
X
Interval of Convergence
How to tell at a glance what the interval of convergence is:
1. If n! appears, ignore everything else. R = 0 or R = ∞.
P∞
2. If not, write your series (if possible) as n=1 A(n) · (x − a)n · rn where rn is the part that increases
exponentially and A(n) is everything else.
3. R = 1/|r|. The interval of convergence is (a − R, a + R), except maybe for the endpoints.
P∞
4. If n=1 A(n) converges absolutely, the interval is [a − R, a + R].
P∞
5. If n=1 A(n) converges conditionally the interval will be one-sided (either (a−R, a+R] or [a−R, a+R)).
6. If limn→∞ A(n) 6= 0, the interval is (a − R, a + R).
Find the intervals of convergence of the following functions:
1.
∞
X
xn
5.
n=1
2.
3.
4.
∞
X
xn
n
n=1
∞
X
6.
∞
X
(x − 5)n
n · 3n
n=1
∞
X
xn
n2
n=1
7.
∞
X
(x − 3)n
2n
n=1
8.
∞
X
n=1
∞
X
n=1
5n
∞
X
n!(x − 1)n
n=1
10.
∞
X
(x − 3)n
n(−4)n
n=1
(x + 3)n
n!
11.
∞
X
xn
n!
n=1
n
2n
12.
(−2)n
n=1
9.
(x + 4)n
√
n
(x + 7)n
1
∞
X
(−1)n
n=1
(x + 2)n
√
n n
Power Series Arithmetic!
1. ex =
3. cos x =
2. sin x =
4.
1
=
1−x
Rewrite the power series above in sigma notation.
Add and Multiply!
1. 5ex =
2. sin x + cos x =
Compose!
1. e−x =
4.
1
=
1 + x2
2. sin(2x) =
5.
1
=
5−x
1
=
1 − 3x
6.
1
=
2 + 3x
2
3.
Differentiate and Integrate!
1.
d x
dx e
2.
d
dx
3.
R
=
sin x =
ex =
R 1
4. ln(1 + x) = 1+x
=
R 1
5. arctan x = 1+x
2 =
R
6. sin x =
. . . Multiply and Divide?
Verify the following:
1. sin(2x) = 2 sin x cos x
2. ex e−x = 1
3. sin2 x + cos2 x = 1
4. Find 1/ cos(x) by solving (a0 + a1 x + a2 x2 + . . .) cos(x) = 1 term-by-term.
Invert??
1. Find arcsin(x) by solving P (sin x) term-by-term.
√
2. Find 1 + x by solving P (x)2 = 1 + x.
2