   SEARCH HOME Math Central Quandaries & Queries  Question from Rahul: Whether the following sequence is increasing or decreasing? I even do not kow whether to call it a sequence or not. an = (1/(1+n))+(1/(2+n))+....+(1/(n+n)). I am confused. It looks like a series to me. Please help. Regards, Rahul Rahul,

It is not a series, because a_n is not obtained by adding one more explicit term to $a_(n-1).$

For instance:

$a_1 = 1/(1+1) = 1/2$

$a_2 = 1/(1+2) + 1/(2+2) = 1/3 + 1/4$

$a_3 = 1/(1+3) + 1/(2+3) + 1/(3+3) = 1/4 + 1/5 + 1/6$

$a_4 = 1/(1+4) + 1/(2+4) + 1/(3+4) + 1/(4+4) = 1/5 + 1/6 + 1/7 + 1/8$

If it were a series then (eg) $a_4$ would be $a_3$ with exactly one new term added on and no other changes - but it isn't.

To guess whether it is increasing or decreasing, evaluate these terms.

To understand it, experiment: write out (say) $a_9$ and $a_{10}.$

Then find and simplify $a_{10} - a_9.$

To prove it, do the same thing using $a_n$ and $a_{n+1}.$

Good Hunting!
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