71I think here we shall use the concept of limits.. \lim_{n\rightarrow \infty } \frac{1}{1+n}
But I don't know how it will be answering your question
Show that 1/2 + 1/3 + 1/4 + ..................... + 1/n can never be an integer. That is, it will always be a fraction.
At last i reached
∑1/(1 + n)
But how to calculate this?
Please help....
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8 Answers
71
21@ Rahul : there is no closed formula for H.P
take lcm ,do the addition, you will get oddeven ,so cant ve an integer , :)
71That's why the poster of solution didn't go any ahead! :P
36It was from yahoo answers........... and he got 10 votes for that.........
If i ever meet him, i'll truly shoot him..... '} - - - ///
21well the solution is quite simple
lets recall how to sum fractions :D
\frac{1}{2}+ \frac{1}{3} + .... + \frac{1}{2^k}+ \frac{1}{2^k +1}+....
where 2k is the highest power of two among number in denominators
SUMMATION : lets take lcm. the lcm will contain (2k)* p1p2...
where p1p2... are all primes
Now ,
\frac{1}{2}+ \frac{1}{3} + .... + \frac{1}{2^k}+ \frac{1}{2^k +1}+..
= \frac{even + even + ... + odd + even + ..}{2^k p_{1}p_{2}....}
[while summing 2k there remains product of primes only , which is necessarily odd]
= \frac{odd}{even}
so not an integer