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Practice Problem
Related Topics
Wize University Calculus 2 Textbook > Sequences and Series
Absolute Convergence & Alternating Series Test
4 Activities
Determine whether the following series are absolutely convergent, conditionally convergent, or divergent:
a.)
∑
n
=
1
∞
(
−
1
)
n
n
\sum\limits^\infty_{n=1}\frac{(-1)^n}{n}
n
=
1
∑
∞
n
(
−
1
)
n
b.)
∑
n
=
0
∞
(
−
2
)
n
\sum\limits^\infty_{n=0}(-2)^n
n
=
0
∑
∞
(
−
2
)
n
c.)
∑
n
=
1
∞
(
−
1
)
n
n
5
3
\sum\limits^\infty_{n=1}\frac{(-1)^n}{\sqrt[3]{n^5}}
n
=
1
∑
∞
3
n
5
(
−
1
)
n
a.) converges conditionally
b.) diverges
c.) converges conditionally
a.) converges absolutely
b.) diverges
c.) converges absolutely
a.) converges conditionally
b.) diverges
c.) converges absolutely
a.) converges absolutely
b.) converges absolutely
c.) converges conditionally
a.) diverges
b.) diverges
c.) converges absolutely
I don't know
Check Submission
More Absolute Convergence & Alternating Series Test Questions:
Practice: Alternating Series Test
Is the series
∑
n
=
2
∞
(
−
1
)
n
+
1
n
2
n
3
+
3
\sum\limits^\infty_{n=2}(-1)^{n+1}\frac{n^2}{n^3+3}
n
=
2
∑
∞
(
−
1
)
n
+
1
n
3
+
3
n
2
convergent or divergent?
Practice: Alternating Series Test
Is the series
∑
n
=
2
∞
(
−
1
)
n
+
1
n
2
n
3
+
3
\sum\limits^\infty_{n=2}(-1)^{n+1}\frac{n^2}{n^3+3}
n
=
2
∑
∞
(
−
1
)
n
+
1
n
3
+
3
n
2
convergent or divergent?
Practice Problem
Example: Absolute Convergence Test
Determine whether the following series are absolutely convergent, conditionally convergent, or divergent:
a.)
∑
n
=
1
∞
(
−
1
)
n
n
\sum\limits^\infty_{n=1}\frac{(-1)^n}{n}
n
=
1
∑
∞
n
(
−
1
)
n
Practice: Alternating Series Test
Practice: AST
Is the series
∑
n
=
2
∞
(
−
1
)
n
+
1
n
2
n
3
+
3
\sum\limits^\infty_{n=2}(-1)^{n+1}\frac{n^2}{n^3+3}
n
=
2
∑
∞
(
−
1
)
n
+
1
n
3
+
3
n
2
convergent or divergent?
Practice: Alternating Series Test
Practice: AST
Is the series
∑
n
=
2
∞
(
−
1
)
n
+
1
n
2
n
3
+
3
\sum\limits^\infty_{n=2}(-1)^{n+1}\frac{n^2}{n^3+3}
n
=
2
∑
∞
(
−
1
)
n
+
1
n
3
+
3
n
2
convergent or divergent?
Practice: Alternating Series Test
Practice: AST
Is the series
∑
n
=
2
∞
(
−
1
)
n
+
1
n
2
n
3
+
3
\sum\limits^\infty_{n=2}(-1)^{n+1}\frac{n^2}{n^3+3}
n
=
2
∑
∞
(
−
1
)
n
+
1
n
3
+
3
n
2
convergent or divergent?
Practice: Alternating Series Test
Practice: AST
Is the series
∑
n
=
2
∞
(
−
1
)
n
+
1
n
2
n
3
+
3
\sum\limits^\infty_{n=2}(-1)^{n+1}\frac{n^2}{n^3+3}
n
=
2
∑
∞
(
−
1
)
n
+
1
n
3
+
3
n
2
convergent or divergent?
Practice Problem
Example: Absolute Convergence Test
Determine whether the following series are absolutely convergent, conditionally convergent, or divergent:
a.)
∑
n
=
1
∞
(
−
1
)
n
n
\sum\limits^\infty_{n=1}\frac{(-1)^n}{n}
n
=
1
∑
∞
n
(
−
1
)
n
Practice: Alternating Series Test
Is the series
∑
n
=
2
∞
(
−
1
)
n
+
1
n
2
n
3
+
3
\sum\limits^\infty_{n=2}(-1)^{n+1}\frac{n^2}{n^3+3}
n
=
2
∑
∞
(
−
1
)
n
+
1
n
3
+
3
n
2
convergent or divergent? If convergent, determine whether it converges conditionally or absolutely.
Practice: Alternating Series Test
Practice: AST
Is the series
∑
n
=
2
∞
(
−
1
)
n
+
1
n
2
n
3
+
3
\sum\limits^\infty_{n=2}(-1)^{n+1}\frac{n^2}{n^3+3}
n
=
2
∑
∞
(
−
1
)
n
+
1
n
3
+
3
n
2
convergent or divergent?
Practice Problem
Example: Absolute Convergence Test
Determine whether the following series are absolutely convergent, conditionally convergent, or divergent:
a.)
∑
n
=
1
∞
(
−
1
)
n
n
\sum\limits^\infty_{n=1}\frac{(-1)^n}{n}
n
=
1
∑
∞
n
(
−
1
)
n