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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- gauravragtah
- I'm Gaurav and I'm building AI for smarter technical hiring (www.profillic.com). I've been living the Silicon Valley life for some years now, I'm originally from New Delhi and now I live in New York. https://www.linkedin.com/in/gauravragtah
2 comments:
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?
Won't the series eventually converge? What makes it different from, say, 1 + .5 + .25 etc...which converges to 2?
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