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By a direct calculation, it is easy to deduce that the simple finite sum $$\sum\limits_{j = 1}^n {\frac{{{r^2}}} {{{j^2}r + j{r^2}}}} = {H_n} + {H_r} - {H_{n + r}},$$ where $H_n$ denotes the harmonic number defined by $${H_n} = \sum\limits_{j= 1}^n {\frac{1}{j}} .$$ For complicated sum $$\sum\limits_{j = 1}^n {\frac{{{r^3}}} {{{j^3}r + {j^2}{r^2} + j{r^3}}}} = ?$$ $$\sum\limits_{j = 1}^n {\frac{{{r^4}}} {{{j^4}r + {j^3}{r^2} + {j^2}{r^3} + j{r^4}}}} = ?$$ They can be expressed in terms of harmonic number?

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  • the use of powers of $i$ is a bit confuse, because $i$ represent, in general, de imaginary number. About the question: decompose the rational function recursively in partial fractions and see what you can get from it. – Masacroso Jul 27 '17 at 09:54
  • Why don't you cancel the unnecessary extra factor $r$ in the numerator and denominator of the summand? – John Bentin Jul 27 '17 at 10:41

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