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Sunday, November 2, 2008

Q) Find the sum of :
1 + 1/2 + 1/3 + 1/4 .... up to infinite terms.

A)
1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + ...
= ( 1/1 ) + ( 1/2 ) + ( 1/3 + 1/4 ) + ( 1/5 + 1/6 + 1/7 + 1/8 ) + ...

(where each set of parenthesis stops at a reciprocal of a power of two.) Replacing the terms in each set of parenthesis by the smallest term (in those parenthesis) we get a smaller sum:

1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + ...
> ( 1/1 ) + ( 1/2 ) + ( 1/4 + 1/4 ) + ( 1/8 + 1/8 + 1/8 + 1/8 ) + ...
= 1/1 + 1/2 + 1/2 + 1/2 + 1/2 + ...

which clearly gets arbitrarily large! So it is a diverging series and will not have a definite value since it is not converging. The sum will go on approaching infinity.

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2 comments:

Light said...

Won't the sequence eventually hit a cap value? What differentiates this sequence from, say, 1 + 1/2 + 1/4 +1/8 etc...which converges to 2?

Light said...

Won't the series eventually converge? What makes it different from, say, 1 + .5 + .25 etc...which converges to 2?