$\large f\left( x \right) =7{ x }^{ 2017 }+{ x }^{ 2 }+x+5$
Let $f\left( x \right)$ be a polynomial having roots ${ x }_{ i }$ for $i=1$ to $2017$ .Then find the value of
$\large \frac { 1 }{ { 10 }^{ 6 } } \left( \sum _{ n=1 }^{ 2017 }{ \prod _{ i=1 }^{ 2017 }{ \left( { x }_{ i }+n \right) } } -\sum _{ k=1 }^{ 2017 }{ { k }^{ 2017 } } \right)$
Details and Assumptions:
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$f(x)$ can be rewritten as: $f(x) = 7\prod\limits_{i = 1}^{2017} {\left( {x - x_i } \right)}$ Now, the sum can be expressed as: $S = \frac{1}{{10^6 }}\left( {\sum\limits_{n = 1}^{2017} {\prod\limits_{i = 1}^{2017} {\left( {x_i + n} \right) - \sum\limits_{k = 1}^{2017} {k^{2017} } } } } \right)$ $S = 10^{ - 6} \left( {\sum\limits_{n = 1}^{2017} {\left( { - 1} \right)^{ - 2017} \left[ {\prod\limits_{i = 1}^{2017} { - \left( {x_i + n} \right)} } \right] - \sum\limits_{k = 1}^{2017} {k^{2017} } } } \right)$ $S = 10^{ - 6} \left( { - \sum\limits_{n = 1}^{2017} {\left[ {\prod\limits_{i = 1}^{2017} {\left( { - n - x_i } \right)} } \right] - \sum\limits_{k = 1}^{2017} {k^{2017} } } } \right)$ $S = - 10^{ - 6} \left( {\sum\limits_{n = 1}^{2017} {\frac{{f\left( { - n} \right)}}{7} + \sum\limits_{n = 1}^{2017} {n^{2017} } } } \right)$ $S = - 10^{ - 6} \left( {\sum\limits_{n = 1}^{2017} { - n^{2017} + \frac{1}{7}n^2 - \frac{1}{7}n + \frac{5}{7} + n^{2017} } } \right)$ $S = \frac{{ - 10^{ - 6} }}{7}\left( {\sum\limits_{n = 1}^{2017} {n^2 - n + 5} } \right)$ $S = \frac{{ - 10^{ - 6} }}{7}\left( {\frac{{2017 \cdot 2018 \cdot 4035}}{6} - \frac{{2017 \cdot 2018}}{2} + 2017 \cdot 5} \right)$ $S = \frac{{ - 2017}}{{84}}\left( {2 \cdot 2018 \cdot 4035 - 6 \cdot 2018 + 60} \right) \cdot 10^{ - 6}$ $S = \frac{{ - 2017}}{{84}}\left( {2018 \cdot 8064 + 60} \right) \cdot 10^{ - 6} \approx \boxed{ - 390.75}$