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Series One

August 31st, 2009

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Put the following numbers into divergent series and a convergent series?

Put the numbers {...,-1/4,-1/3,-1/2,1/2,1/3,1/4,...} into two series: one convergent and one divergent to infinity such that every number is used exactly once in both series. Use partial sums to show the divergence to infinity and the convergence.

Hint: for divergence put blocks of positive numbers together so they sum to at least 1/2.

Convergent:
(1/2 + (-1/2)) + (1/3 + (-1/3)) + ... = 0.
----------------------
Divergent:
1/2 + (-1/2) + (1/3 + 1/4) + (-1/3) + (1/5 + 1/6 + 1/7 + 1/8) + (-1/4) + ...

Why:
1/3 + 1/4 > 1/4 + 1/4 = 1/2
1/5 + 1/6 + 1/7 + 1/8 > 1/8 + 1/8 + 1/8 + 1/8 = 1/2, etc.

So, 1/2 + (-1/2) + (1/3 + 1/4) + (-1/3) + (1/5 + 1/6 + 1/7 + 1/8) + (-1/4) + ...
> 1/2 + (-1/2) + 1/2 + (-1/3) + 1/2 + (-1/4) + ...
= (1/2 - 1/3) + (1/2 - 1/4) + (1/2 - 1/5) + ...
= 1/6 + 2/8 + 3/10 + ... + n/(2(n+2)) + ...

This latter series diverges, because its nth term converges to 1/2, which is nonzero.
Hence, the original series also diverges.

I hope this helps!

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