Arithmetic question

[lionel.tran]

Hi everybody:
I came up with this question and I would like to know what strategy we can use to come up with the answer quickly:
What is the sum of 1/11 + 1/12 + 1/12 + 1/13 + 1/14 + 1/15 + 1/16 + 1/17 + 1/18 + 1/19 + 1/20

Thanks

[lionel.tran]

Hi, can we have help on this question. Is there a formula to derive the answer quickly? thk

[jssaggu.tico]

Hello Sir,

You can't calculate it because it is part of an infinite series and harmonic which doesn't converge.

Harmonic series: The series 1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9+... is called the Harmonic Series, and it sums to infinity (very slowly). In other words, it diverges. Similar series, like 1/2+1/4+1/6+1/8+1/10+... are just different versions of the harmonic series, and diverge. In this one, each term is 1/2 times the original harmonic series.

This series also sums to infinity: 1/2+1/3+1/5+1/7+1/11+1/13+... Here the denominators are prime numbers (see the Sieve Of Eratosthenes). It increases very very slowly, but diverges. How about this series: S=1+1/3+1/5+1/7+1/9+1/11... (with odd denominators)? Well, every term is greater than the corresponding term in the 1/prime series, and so it also diverges. In fact the 1/prime series is included in this series (except for 1/2), another clue that it diverges.

Best Regards

[RonPurewal]

Hello Sir,

You can't calculate it because it is part of an infinite series and harmonic which doesn't converge.

well... not true. you obviously can calculate it - by just adding together all the fractions - but the OP is asking whether there's a more efficient method. unfortunately for the OP, the answer is "no, there is no better way".

as the poster above has noted, these are called "partial harmonic series". if you're interested in finding out more, you can do a search for that particular term - but be warned that you're going to go WAY beyond the scope of the gmat.

--

(it's true that the infinite series does not converge to a finite value - but that's the infinite series. in this problem, we aren't dealing with the infinite series; we're just dealing with ten fractions. of course ten fractions have some sum.)