Exercises in Sequences and Series Convergence, Assignments of Advanced Calculus

A list of exercises on the topic of sequences and series convergence. The exercises cover various types of series, including those with square roots, sine functions, and factorials. Students are asked to determine if the series converge or diverge, and in some cases, whether they are absolutely convergent.

Typology: Assignments

Pre 2010

Uploaded on 07/29/2009

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Exercises #1 (Sequences and Series)
1. For Σ
n=1an= Σ
n=1(1)n1
nand Σ
n=1bn= Σ
n=1(1)n1
n, is the series Σ
n=1anconvergent
or not? Is the series Σ
n=1(an·bn) convergent or not?
2. Find limn→∞
n1+ 1
n
n.
3. Is the series Σ
n=1
1
n1+ 1
n
convergent or divergent? Why?
4. Find limn→∞
nn+1
(n+ 1)n+1 .
5. Is the series Σ
n=1
nn+1
(n+ 1)n+1 convergent or divergent? Why?
6. Is the series Σ
n=1
sin(n2+ 1)
n2+ 1 convergent or divergent? Why?
7. Is the series Σ
n=1
5n+ 5n2
6n+ 4n2convergent or divergent? Why?
8. Is the series Σ
n=1
1
n1+sin n
5
convergent or divergent? Why?
9. Is the series Σ
n=1
1
en2convergent or divergent? Why?
10. Is the series Σ
n=1
2n
·n
3nconvergent or divergent? Why?
11. Is the series Σ
n=1
1
ln(3n21) convergent or divergent? Why?
12. Is the series Σ
n=1
1
n!convergent or divergent? Why?
13. Is the series Σ
n=1(1)n1.0001n
n200 convergent or divergent? Why?
14. Is the series Σ
n=1
n2·2n
n!convergent or divergent? Why?
15. For the series Σ
n=2
(1)n
ln n, is it convergent? Is it absolutely convergent?
16. Is the series Σ
n=1
n!
en2convergent or divergent? Why?
17. Is the series Σ
n=1(n
e1) convergent or divergent? Why? (Hint: Prove first that
limx0
ex
1
x= 1)
1

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Exercises #1 (Sequences and Series)

  1. For Σ∞ n=1an = Σ∞ n=1(−1)n^ √^1 n and Σ∞ n=1bn = Σ∞ n=1(−1)n^ √^1 n , is the series Σ∞ n=1an convergent or not? Is the series Σ∞ n=1(an · bn) convergent or not?
  2. Find limn→∞^ n1+^ n^1 n.
  3. Is the series Σ∞ n=1 n1+^1 n 1 convergent or divergent? Why?
  4. Find limn→∞ (n + 1)^ nn+1n+.
  5. Is the series Σ∞ n=1 (n n+ 1)n+1n+1 convergent or divergent? Why?
  6. Is the series Σ∞ n=1^ sin( nn (^2 2) + 1^ + 1) convergent or divergent? Why?
  7. Is the series Σ∞ n=1^56 nn^ + 5+ 4nn^22 convergent or divergent? Why?
  8. Is the series Σ∞ n=1 n 1+sin^15 n convergent or divergent? Why?
  9. Is the series Σ∞ n=1 e^1 n 2 convergent or divergent? Why?
  10. Is the series Σ∞ n=1^2 n 3 n·^ nconvergent or divergent? Why?
  11. Is the series Σ∞ n=1ln(3n^12 − 1) convergent or divergent? Why?
  12. Is the series Σ∞ n=1 n^1! convergent or divergent? Why?
  13. Is the series Σ∞ n=1(−1)n^1. n^0001200 nconvergent or divergent? Why?
  14. Is the series Σ∞ n=1^ n^2 n·^!^2 nconvergent or divergent? Why?
  15. For the series Σ∞ n=2^ (− ln1) nn , is it convergent? Is it absolutely convergent?
  16. Is the series Σ∞ n=1 e^ nn! 2 convergent or divergent? Why?
  17. Is the series Σ∞ n=1( √ne − 1) convergent or divergent? Why? (Hint: Prove first that limx→ 0 ex^ x− 1 = 1)