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(3n2−1) 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
√e−1) convergent or divergent? Why? (Hint: Prove first that
limx→0
ex
−1
x= 1)
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