APMOPS 2012 Question 13

Find $\frac11+\frac12+\frac22+\frac13+\frac23+\frac33+\frac14+\frac24+\frac34+\frac44+\cdots+\frac1{100}+\frac2{100}+\cdots+\frac{99}{100}+\frac{100}{100}$.