Answer :
In order to find the sum to the indicated accuracy we need to add
5 terms as the series is said to be convergent.
Given Series is Σ(-1/3)^n / n where error is less than 0.0005
= Σ (-1)^n / n3^n
For this series, we will use ratio test
if [tex]\lim_{n \to \infty} |an + 1 / an |[/tex] = L < 1 then series in convergent,
Let an = (-1)^n / n3^n , an+1 = (-1)^n+1 / (n+1)3^n+1
[tex]\lim_{n \to \infty} |an + 1 / an |[/tex] = [tex]\lim_{n \to \infty} | ((-1)^n+1 / (n+1)3^n+1)(n3^n/(-1)^n) |[/tex]
By solving we get,
[tex]\lim_{n \to \infty} |an + 1 / an |[/tex] = 1/3 < 1
Then given series is convergent.
By alternating series estimation theorem we have,
|Rn|=|S-Sn| <= an+1
an+1 <= 0.0005
1 / (n+1)3^n+1 <= 0.0005
(n+1)3^n+1 >= 2000
2 * [tex]3^{2}[/tex] >[tex]\neq[/tex] 2000 (n=1)
4 * [tex]3^{4}[/tex] >[tex]\neq[/tex] 2000 (n=3)
6 * [tex]3^{6}[/tex] >= 200 (n=5)
So required terms is n=5
Learn more about Ratio test :
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