求1+1/2+1/3+1/4+....+1/25的整數部分?

The sum of the first few terms (n) of the series is given analytically by the nth harmonic number.
See:
http://mathworld.wolfram.com/HarmonicSeries.html

For n=25, it is quite simple to sum them numerically (brute force).
The exact sum is 34052522467/8923714800 which is approximately 3.816.
The integer part is therefore 3.

Alternatively, set
S=1+1/2+1/3+....1/25
and we define
S<T=1+4(1/3+1/7+1/11+1/15+1/19+1/23)=3.9195
S>U=1+6(1/4.5+1/10.5+1/16.5+1/22.5)=3.5351
Therefore the integer part of S=3