Question 28
For a real number $x$, $[x]$ is the greatest integer not exceeding $x$ (the integer part). The fractional or decimal part is $\{x\}=x-[x]$, so $x=[x]+\{x\}$. Examples: $[3]=3$, $[6.25]=6$, $[-6.25]=-7$, $[22/7]=3$. The source lists: (1) $0<\{x\}<1$ for all real $x$; (2) $\{x\}=0$ exactly when $x$ is an integer; (3) $x-1<[x]\le x<[x]+1$; (4) $[n+x]=n+[x]$ for positive integer $n$ and real $x$; (5) $[-x]=-[x]-1$ if $x$ is not an integer, and $[-x]=-[x]$ if $x$ is an integer. Source correction: property (1) is printed as $0<\{x\}<1$ for all real $x$, contradicting both the definition and property (2) at integers. The correct bound is $0\le\{x\}<1$. The solution uses the definition and this corrected bound.
Find $[x]$ for $x=\dfrac1{1\cdot2}+\dfrac1{2\cdot3}+\dfrac1{3\cdot4}+\cdots+\dfrac1{n(n+1)}$, where the sum has $n$ terms.
Find $[x]$ for $x=\dfrac1{1\cdot2}+\dfrac1{2\cdot3}+\dfrac1{3\cdot4}+\cdots+\dfrac1{n(n+1)}$, where the sum has $n$ terms.