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MAT136H1 power series
创建日期：2024-05-02 08:46:51
所属分类：数学mathematics
标签： MAT136H1
power series

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MAT136H1

Assignment Week11 due 04/03/2022 at 06:00pm EDT
1. (1 point)
Use the ratio test to find the radius of convergence of the
power series
∞
∑
n=0
(n+2)xn
2n +n
R =
(If the radius is infinite, enter Inf for R.)
Answer(s) submitted:
•
(incorrect)
2. (1 point)
Use the ratio test to find the radius of convergence of the
power series
5x+22 ·5x2 +32 ·5x3 +42 ·5x4 +52 ·5x5 + · · ·
R =
(If the radius is infinite, enter Inf for R.)
Answer(s) submitted:
•
(incorrect)
3. (1 point)
Use the ratio test to find the radius of convergence of the
power series
1+4!x+
8!x2
(2!)4
+
12!x3
(3!)4
+
16!x4
(4!)4
+
20!x5
(5!)4
+ · · ·
R =
(If the radius is infinite, enter Inf for R.)
Answer(s) submitted:
•
(incorrect)
4. (1 point)
If ∑Cn(x− 5)n converges at x = 11 and diverges at x = 14,
what can you say about:
(a) the convergence at x = 17?
• ?
• the series converges
• the series diverges
• the series might converge or might diverge
(b) the convergence at x = 13?
• ?
• the series converges
• the series diverges
• the series might converge or might diverge
(c) the convergence at x = 10?
• ?
• the series converges
• the series diverges
• the series might converge or might diverge
(d) the convergence at x = 0?
• ?
• the series converges
• the series diverges
• the series might converge or might diverge
Answer(s) submitted:
•
•
•
•
(incorrect)
5. (1 point)
The function f (x) = ln(10− x) is represented as a power
series
f (x) =
∞
∑
n=0
cnxn.
Find the first few coefficients in the power series.
c0 =
c1 =
c2 =
c3 =
c4 =
Find the radius of convergence R of the series.
R = .
Answer(s) submitted:
•
•
•
•
•
•
(incorrect)
1
6. (1 point)
Suppose that you are told that the Taylor series of f (x) =
x3ex
3
about x = 0 is
x3 + x6 +
x9
2!
+
x12
3!
+
x15
4!
+ · · · .
Find each of the following:
d
dx
(
x3ex
3
)∣∣∣∣
x=0
=
d9
dx9
(
x3ex
3
)∣∣∣∣
x=0
=
Answer(s) submitted:
•
•
(incorrect)
7. (1 point)
Consider the functions y = 1− ln(1+ x2) and y = cos(x).
A. Write the Taylor expansions for the two functions about
x = 0. What is similar about the two series? What is different?
Looking at the series, which function do you predict will be
greater over the interval (-1,1)?
• A. 1− ln(1+ x2)
• B. cos(x)
(Graph the functions to verify that your answer is correct!)
B.
Are these functions even or odd?
• A. Even
• B. Odd
C. Find the radii of convergence for your two series.
For 1− ln(1+ x2), the radius of convergence is
For cos(x), the radius of convergence is
(Enter infinity if the radius of convergence is infinite.)
Looking at the relative sizes of the successive terms in your
series, note how the radii of convergence you found make sense.
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