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[parent] proof of alternating series test (Proof)

If the first term $ a_1$ is positive, then the series has partial sum

$\displaystyle S_{2n+2}=a_1-a_2+a_3+...-a_{2n}+a_{2n+1}-a_{2n+2}, $
where the $ a_j$'s are all nonnegative and nonincreasing. (If the first term is negative, consider the series in the absence of the first term.) From above, we have the following:
$\displaystyle S_{2n+1}$ $\displaystyle =S_{2n}+a_{2n+1};$    
$\displaystyle S_{2n+2}$ $\displaystyle =S_{2n}+(a_{2n+1}-a_{2n+2});$    
$\displaystyle S_{2n+3}$ $\displaystyle =S_{2n+1}-(a_{2n+2}-a_{2n+3})$    
  $\displaystyle =S_{2n+2}+a_{2n+3}.$    

Since $ a_{2n+1} \geq a_{2n+2}\geq a_{2n+3}$, we have $ S_{2n+1}\geq S_{2n+3} \geq S_{2n+2} \geq S_{2n}$. Moreover,

$\displaystyle S_{2n+2}=a_1-(a_2-a_3)-(a_4-a_5)-\cdots-(a_{2n}-a_{2n+1})-a_{2n+2}. $
Because the $ a_j$'s are nonincreasing, we have $ S_n \geq 0$ for any $ n$. Also, $ S_{2n+2} \leq S_{2n+1} \leq a_1$. Thus, $ a_1 \geq S_{2n+1} \geq S_{2n+3} \geq S_{2n+2} \geq S_{2n} \geq 0$. Hence, the even partial sums $ S_{2n}$ and the odd partial sums $ S_{2n+1}$ are bounded. Also, the even partial sums $ S_{2n}$'s are monotonically nondecreasing, while the odd partial sums $ S_{2n+1}$'s are monotonically nonincreasing. Thus, the even and odd series both converge.

We note that $ S_{2n+1}-S_{2n}=a_{2n+1}$. Therefore, the sums converge to the same limit if and only if $ a_n\to 0$ as $ n\to\infty$. The theorem is then established.



"proof of alternating series test" is owned by Wkbj79. [ full author list (2) | owner history (2) ]
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Cross-references: limit, sums, converge, monotonically nondecreasing, bounded, odd, even, negative, nonincreasing, partial sum, series, positive, term
There is 1 reference to this entry.

This is version 9 of proof of alternating series test, born on 2002-05-24, modified 2007-10-05.
Object id is 2936, canonical name is ProofOfAlternatingSeriesTest2.
Accessed 4651 times total.

Classification:
AMS MSC: 40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)
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