Calculus II Activity 6 – Chapter 11
1.
𝑎𝑎𝑛𝑛 =
2.
∞
�
𝑛𝑛=1
3.
∞
�
𝑛𝑛=2
Determine whether the sequence is convergent or divergent. If it is convergent, �ind its limit.
𝑛𝑛 sin 𝑛𝑛
𝑛𝑛2 + 1
Determine whether the series is convergent or divergent.
5𝑛𝑛4 + 3𝑛𝑛3 + 𝑛𝑛
6𝑛𝑛4 + 4𝑛𝑛 + 2
Determine whether the series is convergent or divergent. If it is convergent, �ind its sum.
4
𝑛𝑛2 + 𝑛𝑛
4.
∞
�
𝑛𝑛=2
5.
∞
�
𝑛𝑛=1
ln 𝑛𝑛
𝑛𝑛3
Use the Integral Test to determine whether the series is convergent or divergent.
Determine whether the series is convergent or divergent.
2𝑛𝑛2 + 1
3𝑛𝑛3 + 1
6.
∞
Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
�(−1)𝑛𝑛 𝑛𝑛−1/4
𝑛𝑛=1
7.
∞
�
𝑛𝑛=1
Determine whether the series is convergent or divergent.
(−7)2𝑛𝑛
𝑛𝑛2 16𝑛𝑛
8.
∞
�
𝑛𝑛=1
9.
∞
�
𝑛𝑛=1
Determine whether the series is convergent or divergent.
1 ∙ 3 ∙ 5 ⋯ (2𝑛𝑛 − 1)
3𝑛𝑛 𝑛𝑛!
Find the radius of convergence and interval of convergence of the power series.
32𝑛𝑛 𝑛𝑛
𝑥𝑥
𝑛𝑛
10.
𝑓𝑓(𝑥𝑥) =
Find a power series representation for the function and determine the interval of convergence.
2 + 𝑥𝑥
(1 − 𝑥𝑥)2
11.
Find the Maclaurin series for 𝑓𝑓 using the de�inition of a Maclaurin series. Find the associated
radius of convergence.
𝑓𝑓(𝑥𝑥) = 𝑒𝑒 −3𝑥𝑥
12.
Find the Taylor series for 𝑓𝑓 centered at the given value of 𝑎𝑎. Find the associated radius of
convergence.
𝑓𝑓(𝑥𝑥) = ln 𝑥𝑥 , 𝑎𝑎 = 3
13.
∞
Find the function represented by the given power series.
3𝑛𝑛
(−1)𝑛𝑛 𝑥𝑥 2
�
𝑛𝑛!
𝑛𝑛=0