Number of roots of <msubsup> <mi mathvariant="normal">&#x03A3;<!-- Σ --> <mrow class="MJX

Despiniosnt

Despiniosnt

Answered question

2022-05-27

Number of roots of Σ r = 1 2009 r x r

Answer & Explanation

humanistex3

humanistex3

Beginner2022-05-28Added 9 answers

The function has vertical asymptotes at each x = 1 , 2 , 3 , , 2009. At each interval I k = [ k , k + 1 ] , k = 1 , 2008, you have that:
lim x k + f ( x ) = + , lim x ( k + 1 ) f ( x ) = , f ( x ) < 0.
This proves that there is exactly one root at each interval I k , hence 2008 roots in the interval [ 1 , 2009 ]. To complete the proof you must show that there are no roots outside this interval.

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