Alternate series converges

Prove that (1-2) - (1-2^(1/2)) + (1 - 2^(1/3)) - (1 - 2^(1/4)) + ... converges.

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Originally Posted by tarheelborn

Prove that (1-2) - (1-2^(1/2)) + (1 - 2^(1/3)) - (1 - 2^(1/4)) + ... converges.

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Just show monotonically, then just apply the alternating series test.

Ah, so since (1-2) - (1- 2^(1/2))+... = (-1)(2-1) + (1)(2^(1/2) -1)-... = \sum (-1)^n(2^(1/n)-1). Lim 2^(1/n) = 1 ==> Lim 2^(1/n)-1 = 0. Also, 2^(1/n)>=2^1/(n+1)>=2^(1/n+2)... so {2^(1/n)} is nonincreasing. Thus this series is convergent by alternate series test. That will work, right? Thanks!

Quote:

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Originally Posted by tarheelborn

Ah, so since (1-2) - (1- 2^(1/2))+... = (-1)(2-1) + (1)(2^(1/2) -1)-... = \sum (-1)^n(2^(1/n)-1). Lim 2^(1/n) = 1 ==> Lim 2^(1/n)-1 = 0. Also, 2^(1/n)>=2^1/(n+1)>=2^(1/n+2)... so {2^(1/n)} is nonincreasing. Thus this series is convergent by alternate series test. That will work, right? Thanks!

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Correct. You might want to say why .