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Topic: Convergent Series (Read 704 times) |
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ThudnBlunder
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Convergent Series
« on: Apr 1st, 2007, 5:20pm » |
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For what range of a > 0 does the series below converge?
 a(1 + 1/2 + 1/3 + ....... + 1/n)
n=1
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THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
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iyerkri
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Re: Convergent Series
« Reply #1 on: Apr 1st, 2007, 8:42pm » |
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My answer : a < 1/e .
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Icarus
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Re: Convergent Series
« Reply #2 on: Apr 2nd, 2007, 5:05pm » |
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Providing details: Let L(n) = 1 + 1/2 + ... +1/n. Obviously, the series diverges for a  1, so assume a < 1.
ln(n) < L(n) < ln(n) +  , so a aln(n)  aL(n) aln(n). Hence the series converges or diverges with aln(n). But aln(n) = nln(a), and nln(a) is well known to converge if and only if ln(a) < -1.
Hence aL(n) converges if and only if 0 < a < 1/e.
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| « Last Edit: Apr 2nd, 2007, 5:15pm by Icarus » |
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