Wize AP Calculus (BC) Textbook > Sequences and Series

Absolute Convergence & Alternating Series Test

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Absolute Convergence Test
Note: If ∑n=1∞∣an∣\sum\limits^\infty_{n=1}|a_n|n=1∑∞​∣an​∣ converges, then ∑n=1∞an\sum\limits^\infty_{n=1}a_nn=1∑∞​an​ converges.

Wize Concept
Definitions:
We say that a series, ∑n=1∞an\sum\limits^\infty_{n=1}a_nn=1∑∞​an​ is absolutely convergent if the series ∑n=1∞∣an∣\sum\limits^\infty_{n=1}|a_n|n=1∑∞​∣an​∣ converges
We say that a series, ∑n=1∞an\sum\limits^\infty_{n=1}a_nn=1∑∞​an​ is conditionally convergent if it converges but is not absolutely convergent
*If the terms in the series are all positive, we don't have to make this distinction.

Absolute Convergence Test
If it is easier to work with ∣an∣\left|a_n\right|∣an​∣, then determine whether ∑n=1∞∣an∣\sum\limits^\infty_{n=1}|a_n|n=1∑∞​∣an​∣ converges.
If it does converge, then the original series ∑n=1∞an\sum\limits^\infty_{n=1}a_nn=1∑∞​an​ must also converge.

Identifying Clues
If the series involves negative and positive terms
The question will ask specifically for "absolute convergence", "conditional convergence", or divergence

Example: Absolute Convergence Test
Determine whether the following series are absolutely convergent, conditionally convergent, or divergent:

a.) ∑n=1∞(−1)nn\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)^nn=0∑∞​(−2)n

c.) ∑n=1∞(−1)nn53\sum\limits^\infty_{n=1}\frac{(-1)^n}{\sqrt[3]{n^5}}n=1∑∞​3n5​(−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

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Alternating Series Test
Consider a series with terms that have alternating signs ∑n=1∞an=∑n=1∞(−1)n+1bn,\sum\limits^\infty_{n=1}a_n=\sum\limits^\infty_{n=1}(-1)^{n+1}b_n,n=1∑∞​an​=n=1∑∞​(−1)n+1bn​, 
where the terms bnb_nbn​ are positive.
If bn+1≤bnb_{n+1}\le b_nbn+1​≤bn​ (the terms do not increase) after a certain point  and
If lim⁡n→∞bn=0\lim\limits_{n\rightarrow\infty}b_n =0n→∞lim​bn​=0
Then the series converges.


Identifying Clues:
The terms have alternating signs

Practice: AST
Is the series∑n=2∞(−1)n+1n2n3+3\sum\limits^\infty_{n=2}(-1)^{n+1}\frac{n^2}{n^3+3}n=2∑∞​(−1)n+1n3+3n2​ convergent or divergent?