1.
\(\set{\frac{n^3}{n^4+1}}\)
Section 4.2 Series Review
The three main question types for series (and sequences) problems are:
-
Find the limit of the sequence, or determine if the sequence diverges.
-
Determine if series converges or diverges.
-
Find the value of the sum of the series, or determine if it diverges.
-
Determine if series converges absolutely, converges conditionally, or diverges.
Subsection 4.2.1 Exercises
Exercises Exercises
Limit of sequences.
Find the limit of each sequence, or determine if the sequence diverges.
2.
\(\set{\frac{n^{12}}{3n^{12}+4}}\)
3.
\(\set{\frac{3n^3-1}{2n^3+1}}\)
4.
\(\set{\frac{n^5+3n}{10n^3+n}}\)
5.
\(\set{\tan^{-1}\brac{\frac{10n}{10n+4}}}\)
6.
\(\set{1+\cos\brac{\frac{1}{n}}}\)
7.
\(\set{\ln\brac{n^3+1}-\ln\brac{3n^3+10n}}\)
8.
\(\set{\ln\brac{\sin\brac{\frac{1}{n}}}+\ln n}\)
9.
\(\set{\frac{n}{\sqrt{9n^2+1}}}\)
10.
\(\set{\frac{\sqrt{4n^4+3n}}{8n^2+1}}\)
11.
\(\set{\frac{2e^n+1}{e^n}}\)
12.
\(\set{\frac{\ln(n^2)}{\ln(3n)}}\)
13.
\(\set{5(-1.01)^n}\)
14.
\(\set{2^n+3^{-n}}\)
15.
\(\set{100(-0.003)^n}\)
16.
\(\set{(0.5)^n+3(0.75)^n}\)
17.
\(\set{\frac{e^n+\pi^n}{e^n}}\)
18.
\(\set{\frac{3^{n+1}+3}{3^n}}\)
19.
\(\set{\frac{3^n}{3^n+4^n}}\)
20.
\(\set{\frac{(n+1)!}{n!}}\)
21.
\(\set{\frac{(2n)! \cdot n^2}{(2n+2)!}}\)
22.
\(\set{\tan^{-1} n}\)
23.
\(\set{\frac{e^{n/10}}{2^n}}\)
24.
\(\set{\sqrt{n^2+1}-n}\)
25.
\(\set{\frac{n^2}{n}}\)
26.
\(\set{\brac{1+\frac{2}{n}}^n}\)
27.
\(\set{\frac{\ln n}{n^{1.1}}}\)
28.
\(\set{\sqrt[n]{e^{3n+4}}}\)
29.
\(\set{\brac{\frac{n}{n+5}}^n}\)
30.
\(\set{\sqrt{\brac{1+\frac{1}{2n}}^n}}\)
31.
\(\set{\brac{1+\frac{4}{n}}^{3n}}\)
32.
\(\set{\frac{n}{e^n+3n}}\)
33.
\(\set{\frac{\ln\brac{\frac{1}{n}}}{n}}\)
34.
\(\set{\brac{\frac{1}{n}}^{1/n}}\)
35.
\(\set{\brac{1-\frac{4}{n}}^n}\)
36.
\(\set{b_n}, \quad b_n=\begin{cases}
\frac{n}{n+1}, \amp n \leq 5000 \\
ne^{-n}, \amp n \gt 5000
\end{cases}\)
37.
\(\set{\cot\brac{\frac{n\pi}{2n+2}}}\)
38.
\(\set{n\brac{1-\cos\frac{1}{n}}}\)
39.
\(\set{n\sin\frac{6}{n}}\)
40.
\(\set{\frac{n^8+n^7}{n^7+n^8\ln n}}\)
Convergence of series.
Determine if each series converges or diverges.
41.
\(\sum_{n=1}^{\infty} \frac{n^2-1}{n^3+1}\)
42.
\(\sum_{n=2}^{\infty} \frac{1}{n\sqrt{\ln n}}\)
43.
\(\sum_{n=1}^{\infty} \brac{\frac{1}{n^3}+\frac{1}{3^n}}\)
44.
\(\sum_{n=1}^{\infty} \frac{3^n n^2}{n!}\)
45.
\(\sum_{n=1}^{\infty} \frac{\sin(2n)}{1+2^n}\)
46.
\(\sum_{n=1}^{\infty} \frac{n^{2n}}{(1+n)^{3n}}\)
47.
\(\sum_{n=1}^{\infty} \frac{n^{\sqrt{2}}}{2^n}\)
48.
\(\sum_{n=1}^{\infty} \frac{2^{n-1} 3^{n+1}}{n^n}\)
49.
\(\sum_{n=1}^{\infty} \frac{\sqrt{n^4+1}}{n^3+n}\)
50.
\(\sum_{n=1}^{\infty} \frac{1\cdot 3\cdot 5\cdots (2n-1)}{2\cdot 5\cdot 8\cdots (3n-1)}\)
51.
\(\sum_{n=1}^{\infty} \frac{\sqrt[3]{n}-1}{\,n\brac{\sqrt{n}+1}}\)
52.
\(\sum_{n=1}^{\infty} \frac{1}{n \cdot 3^n}\)
53.
\(\sum_{n=1}^{\infty} \brac{\frac{2n+3}{5n+4}}^n\)
54.
\(\sum_{n=2}^{\infty} \frac{1}{\ln n}\)
55.
\(\sum_{n=2}^{\infty} \frac{(\ln n)^2}{n^3}\)
56.
\(\sum_{n=1}^{\infty} \frac{n!}{10^n}\)
57.
\(\sum_{n=1}^{\infty} \frac{n^{10}}{10^n}\)
58.
\(\sum_{n=1}^{\infty} \brac{\frac{n-2}{n}}^n\)
59.
\(\sum_{n=1}^{\infty} \frac{2+(-1)^n}{1.25^n}\)
60.
\(\sum_{n=1}^{\infty} \brac{\frac{-2}{3}}^n\)
61.
\(\sum_{n=1}^{\infty} \brac{1-\frac{1}{3n}}^n\)
62.
\(\sum_{n=1}^{\infty} \frac{\ln n}{n^3}\)
63.
\(\sum_{n=2}^{\infty} \frac{(-\ln n)^n}{n^n}\)
64.
\(\sum_{n=1}^{\infty} \brac{\frac{1}{n}-\frac{1}{n^2}}\)
65.
\(\sum_{n=1}^{\infty} \frac{e^n}{n^e}\)
66.
\(\sum_{n=1}^{\infty} \frac{n\ln n}{(-2)^n}\)
67.
\(\sum_{n=1}^{\infty} \frac{(n+1)(n+2)}{n!}\)
68.
\(\sum_{n=1}^{\infty} e^{-n^3}\)
69.
\(\sum_{n=1}^{\infty} \frac{(n+3)!}{3!\,n!\,3^n}\)
70.
\(\sum_{n=1}^{\infty} \frac{n\,2^n\,(n+1)!}{3^n n!}\)
71.
\(\sum_{n=1}^{\infty} \frac{n!}{(2n+1)!}\)
72.
\(\sum_{n=1}^{\infty} n^2 e^{-2n}\)
73.
\(\sum_{n=1}^{\infty} \frac{n!}{(-n)^n}\)
74.
\(\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^2}\)
75.
\(\sum_{n=3}^{\infty} \frac{\ln n}{\ln(\ln n)}\)
76.
\(\sum_{n=1}^{\infty} \frac{n+1}{n!}\)
77.
\(\sum_{n=1}^{\infty} \brac{\frac{1}{n}-\frac{1}{n^2}}^n\)
78.
\(\sum_{n=1}^{\infty} \frac{2^n 3^n}{n^n}\)
79.
\(\sum_{n=1}^{\infty} \frac{1}{\sqrt{n(n+1)(n+2)}}\)
80.
\(\sum_{n=2}^{\infty} \frac{1}{n\sqrt{n^2-1}}\)
Sum of a series.
Find the value of the sum of each series, or determine if it diverges. Simplify your answer completely.
81.
\(\sum_{n=3}^{\infty} \frac{1}{(2n-3)(2n-1)}\)
82.
\(\sum_{n=1}^{\infty} 3\,(1.001)^n\)
83.
\(\sum_{n=2}^{\infty} \frac{-2}{n(n+1)}\)
84.
\(\sum_{n=1}^{\infty} \brac{\frac{9}{10}}^n\)
85.
\(\sum_{n=1}^{\infty} \frac{9}{(3n-1)(3n+2)}\)
86.
\(\sum_{n=0}^{\infty} \brac{\frac{1}{3}}^n + \brac{\frac{4}{3}}^n\)
87.
\(\sum_{n=1}^{\infty} \frac{1}{n(n+3)}\)
88.
\(\sum_{n=1}^{\infty} \frac{3^n-5^n}{10^n}\)
89.
\(\sum_{n=3}^{\infty} \frac{-8}{(4n-3)(4n+1)}\)
90.
\(\sum_{n=1}^{\infty} \ln\!\brac{\frac{2n+1}{2n-1}}\)
91.
\(\sum_{n=0}^{\infty} e^{-n}\)
92.
\(\sum_{n=0}^{\infty} \brac{-\frac{1}{5}}^n\)
93.
\(\sum_{n=1}^{\infty} (-1)^n \frac{3}{4^n}\)
94.
\(\sum_{n=1}^{\infty} \brac{\tan^{-1}(n+1)-\tan^{-1}n}\)
95.
\(\sum_{n=2}^{\infty} \brac{\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n-1}}}\)
96.
\(\sum_{n=2}^{\infty} \brac{\frac{2^{n+1}}{3^n} + \frac{1}{2n-1} - \frac{1}{2n+1}}\)
97.
\(\sum_{n=1}^{\infty} \frac{9}{(3n-2)(3n+1)}\)
98.
\(\sum_{n=1}^{\infty} 4^{-3n}\)
99.
\(\sum_{n=1}^{\infty} \frac{2^n}{3^{n+2}}\)
Absolute convergence and conditional convergence.
Determine if each series converges absolutely, converges conditionally, or diverges.
100.
\(\sum_{n=1}^{\infty} \frac{(-1)^{n+1} n^2}{n^3+1}\)
101.
\(\sum_{n=1}^{\infty} \frac{(-1)^n 10^n}{(n+1)!}\)
102.
\(\sum_{n=1}^{\infty} (-5)^{-n}\)
103.
\(\sum_{n=1}^{\infty} (-1)^n \frac{n^2-1}{n^3+1}\)
104.
\(\sum_{n=1}^{\infty} n!(-e)^{-n}\)
105.
\(\sum_{n=1}^{\infty} (-1)^n \frac{n^2-1}{n^2+1}\)
106.
\(\sum_{n=1}^{\infty} (-1)^{n-1}\frac{n^4}{4^n}\)
107.
\(\sum_{n=1}^{\infty} (-1)^{n+1}\left(\frac{n}{10}\right)^n\)
108.
\(\sum_{n=1}^{\infty} \frac{(-1)^n}{\ln(n+1)}\)
109.
\(\sum_{n=1}^{\infty} \frac{(-1)^n}{n\sqrt{n^2+1}}\)
110.
\(\sum_{n=1}^{\infty} \frac{(-1)^n 3n^2}{n^3+1}\)
111.
\(\sum_{n=1}^{\infty} \frac{(-1)^n (n^2+1)}{2n^2+n-1}\)
112.
\(\sum_{n=1}^{\infty} (-1)^n n^2 e^{-n}\)
113.
\(\sum_{n=1}^{\infty} (-1)^n \brac{1-\frac{3}{n}}^n\)
114.
\(\sum_{n=2}^{\infty} (-1)^{n-1}\frac{1}{\sqrt{n}-1}\)
