Lecture Notes of Thomas' Calculus Chapter 8 Exercises

Lecture Notes of
Thomas’ Calculus
Chapter 8
Exercises
Instructor: Dr. ZHANG Zhengru

Lecture Notes of Chapter 9-Exercises Page 1
OUTLINE
• Series and Partial Sums
• Geometric Series
• Divergence Series
• nth-Term Test for Divergence
• Adding or Deleting Terms
• Reindexing
• Combing Series
EXERCISES
1. Determine convergence or divergence
(a)
(b)
(c)
(d)
∞∑
n=1
∞∑
n=1
(n!) 2
2n 2
diverges
n cos nπ
3
2 n converges absolutely
∞∑
(−1) n n ln n + 1 converges conditionally
n
∞∑
(
sin 1 )
2n − sin 1
converges absolutely
2n + 1
n=1
n=1
2. Find the sum
(a)
(b)
(c)
∞∑
n=1
∞∑
n=1
∞∑
n=1
x n
n(n + 1)
⎧
⎨ 1 + ( 1 − 1) ln(1 − x) x ∈ (−1, 0) ∪ (0, 1)
x
s(x) =
⎩ 0 x = 0
n 2
n!2 (Hint: consider s(x) = ∑ ∞
n 2
n n! xn s( 1 2 ) = 3 √ e)
4
n=1
2n − 1
2 n (Hint: consider (s(x) =
∞∑
n=1
2n − 1
2 n x 2n−2 s(1) = 3)

Lecture Notes of Chapter 9-Exercises Page 2
(d) π − π3
3! + π5
5! − · · · + (−1)n π 2n+1
2(n + 1)! + · · ·
(d) 1 −
(f) 1 + ln 2 +
π2
9 · 2! + π4
81 · 4! − · · ·
(ln 2)2
2!
+ · · · + (ln2)n
n!
+ · · ·
3. Find Fourier Sine and Cosine series for
⎧
⎨ 1 0 ≤ x ≤ h
f(x) =
⎩ 0 h < x ≤ π
4. Find the following sums
σ = 1 + 1 2 2 + 1 3 2 + 1 4 2 · · ·
σ 1 = 1 + 1 3 2 + 1 5 2 + 1 7 2 · · ·
σ 2 = 1 2 2 + 1 4 2 + 1 6 2 · · ·
σ 3 = 1 − 1 2 2 + 1 3 2 − 1 4 2 · · ·
Solution:
Consider the Fourier series generating by the following function
⎧
⎨ −x −π ≤ x < 0
f(x) =
⎩ x 0 ≤ x ≤ π
The Fourier coefficients are as following
⎧
a n = 1 ∫ π
⎨ − 4 n = 1, 3, 5, · · ·
n
f(x) cosnxdx =
2 π
π −π
⎩ 0 n = 2, 4, 6, · · ·
a 0 = 1 π
∫ π
−π
b n = 0
f(x)dx = π
Thus
f(x) = π 2 − 4 π (cosx + 1 3 cos 3x + 1 cos 5x + · · ·)
3 52 x = 0, f(0) = 0 then
σ 1 = π2
8 = 1 + 1 3 3 + 1 5 2 + 1 7 2 · · ·

Lecture Notes of Chapter 9-Exercises Page 3
Note that
σ 2 = 1 4 σ = 1 4 (σ 1 + σ 2 )
so σ 2 = σ 1
3 = π2
24 .
Finally, σ = σ 1 + σ 2 = π2
6 and σ 3 = σ 1 − σ 2 = π2
12
5. Find the limit
√
√
2, 2 + √ √
√
2, 2 + 2 + √ 2, · · ·
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