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NastyAccident
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Homework Statement
Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy?
[tex]\sum^{\infty}_{n=1}\frac{(-1)^{n}}{n*9^{n}}[/tex]
(|error| < 0.0001)
Homework Equations
Alternating Series Test
General knowledge of adding series up...
The Attempt at a Solution
Alternating Series Test
1.) Limit x->infinity
[tex]\frac{(-1)^{n}}{n*9^{n}}[/tex] = 0
2.) Decreasing eventually for n (really immediately)
[tex]\sum^{\infty}_{n=1}\frac{(-1)^{n+1}}{(n+1)*9^{n+1}} < \sum^{\infty}_{n=1}\frac{(-1)^{n}}{n*9^{n}}[/tex]
By the Alternating Series Test, this series ([tex]\sum^{\infty}_{n=1}\frac{(-1)^{n}}{n*9^{n}}[/tex]) is convergent.
Well, I originally choose n = 4 terms in order to find the indicated accuracy. However, that was wrong.
So, I'm sort of scratching my head as to what I did wrong...
My approach:
[tex]\sum^{\infty}_{n=1}\frac{(-1)^{n}}{n*9^{n}}[/tex]
I start listing out an's terms -
|an| < 0.0001
a1 = 0.11111111111
a2 = 0.006172839506
a3 = 0.000457247370
a4 = 0.000038103947
a5 = 0.000003387017
a6 = 0.000000313612
a7 = 0.000000029867
a4 < 0.0001, so n should equal four terms... However, it doesn't for some odd reason?
Any help/suggestions as to where I went wrong are appreciated and will be thanked!
NastyAccident
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