These are practice problems for series and power series. For all of the
following series check (and fully justify why) whether the series converges
or diverges. For problems with a star, if the series converges find its sum.
If the problem is about a power series find both its radius and interval of
convergence
1.
∞
X
2 ln n
n=1
2.
∞
X
cos2 (5n)
n=1
3.
4.
5.
∞
X
7
n5n
n=1
∞
X
∞
X
8.
n3
n3 2n
n!
∞
X
8n
32n
n=1
∞
X
n=1
9.
4+
1
√
4
(−1)n
n=1
7.
n3
∞
X
(−4)n
n n
6
n=1
n=1
6.
n6
2n2
9n2 − 7
∞
X
10 + 9n
n=1
1
5 + 8n
10.
11.
∞
X
6n
6n+4
n=1
∞
X
9n + 2n
12.
∞
X
(−1)n
n=1
13.
[∗]
10n
n=1
∞
X
n3 + 1
n3 − 7
n!(x − 4)n
n=1
14.
∞
X
(−1)n n7n xn
n=1
15.
16.
∞
X
xn (n + 9)
(−1)n
5n
n=1
∞
X
(−1)n
n=1
cos n
n5
Hint: it converges; a bit tricky, you need to apply two tests one after
the other
2