In exercise 1, find a defining formula an = f (n) for the sequence.
1.
1 2 22 23 24
1)−4,−3,−2,−1,0,··· 2) ,− , ,− , ,···
9 12 15 18 21
In exercise 2-6, determine the convergence or divergence of the sequences. If the sequence is convergent, find the limit.
2.
(1) an = 1+(−1)n
n+1 1
(2) an = 1−
2n n
3.
sin2(2n+1)
(1) an = 2
n
cos(2n+3)
(2) an = n
2
4.
n+(−1)n+1
(1) an = 2n
5.
2n+1
(2) an = √
1−3 n
ln(2n+1)
(1) an = √
n
1
(2) an = cos(2π + 2) n
6.
(−4)n
(1) an = n!
1
(2) an = 2+( )2n
2
7. Determine if the geometric series converges or diverges. If the series converges, find
the value. ∞ ∞
X (−1)n X (−3)n
(1) 4n+1 (2) 2n n=1 n=1
8. Find a formula for the n-th partial sume of the series and use it to determine if the series converges or diverges. If a series converges, find its value.
∞ 3 3 ! X∞ √ √
X
(1) n2 − (n+1)2 (2) n+4− n+3 n=1 n=1
