1/2+2/3+3/4+4/5......+2013/2014 =
That has 2013 terms, and is equal to
(1-1/2)+(1-1/3)+(1-1/4)+(1-1/5)+...+(1-1/2014) =
(1+1+...+1)-1/2-1/3-1/4-1/5-...-1/2014) =
There are 2013 1's added so this is:
2013 - (1/2+1/3+1/4+1/5+...+1/2014)
That series in parentheses starts with 1/2.
The harmonic series starts with 1/1, so let's
add and subtract 1:
which equals:
2013 + 1 - 1 - (1/2+1/3+1/4+1/5+...+1/2014) =
2014 - (1+1/2+1/3+1/4+1/5+...+1/2014)
In the parentheses in the harmonic sequence to 2014 terms.
There is no formula for that, but a good approximation for it is:
Hn = ln(n) + g + 1/(2n) - 1/(12n^2)
where g = the Euler-Mascheroni constant, which
is approximately 0.5772156649. You can Google information on that.
Substituting n = 2014
H2014 is approximately 8.185342021, making the desired sum
approximately
2014 - 8.185342021 = 2005.814658
When I ran that sum on the computer, I got 2005.81466
Edwin