What is Alternating series: Definition and 109 Discussions

In mathematics, an alternating series is an infinite series of the form





∑

n
=
0


∞


(
−
1

)

n



a

n




{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}a_{n}}
or




∑

n
=
0


∞


(
−
1

)

n
+
1



a

n




{\displaystyle \sum _{n=0}^{\infty }(-1)^{n+1}a_{n}}
with an > 0 for all n. The signs of the general terms alternate between positive and negative. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges.

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  2. karush

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  5. Mr Davis 97

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  9. B

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  10. H

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  11. Drakkith

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  12. I

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  13. M

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  15. J

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  16. M

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  17. A

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  19. J

    Calc II - Alternating Series Test/Limits

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  23. J

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  24. F

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  27. C

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  28. J

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  29. F

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  31. T

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  32. W

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  33. H

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  39. D

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    1 \ - \ \frac{1}{2} \ + \ \frac{1}{3} \ - \ \frac{1}{4} \ + \ ... \ - \ \frac{1}{n - 1} \ + \ \frac{1}{n} \ - \ \frac{1}{2n + 1} \ < \ ln(n), where n is a positive odd integer I worked this out (rediscovered it) and proved it by induction. For example, when n = 71...
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