proof that Σ_(i=1) ^n (a_i /(a_i −x))=2015 has exactly n real roots.o<a_1 ....<a


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Question Number 55685 by tm888 last updated on 02/Mar/19

$p r o o f t h a t$ $\underset{i = 1}{\sum^{n}} \frac{a_{i}}{a_{i} - x} = 2015 h a s e x a c t l y n r e a l$ $r o o t s . o < a_{1} . . . . < a_{n}$
	Answered by mr W last updated on 03/Mar/19

	$l e t y = f (x) = \underset{i = 1}{\sum^{n}} \frac{a_{i}}{a_{i} - x}$ $y^{'} = \underset{i = 1}{\sum^{n}} \frac{a_{i}}{{(a_{i} - x)}^{2}} > 0$ $\Rightarrow t h e f u n c t i o n i s s t r i c k l y i n c r e a s i n g .$ $\lim_{x \to a_{k}^{(-)}} f (x) = \underset{i = 1}{\sum^{n}} [\lim_{x \to a_{k}^{(-)}} \frac{a_{i}}{a_{i} - x}]$ $= \underset{i = 1, \neq k}{\sum^{n}} \frac{a_{i}}{a_{i} - a_{k}} + \lim_{x \to a_{k}^{(-)}} \frac{a_{k}}{a_{k} - x} = + \infty$ $\lim_{x \to a_{k}^{(+)}} f (x) = \underset{i = 1}{\sum^{n}} [\lim_{x \to a_{k}^{(+)}} \frac{a_{i}}{a_{i} - x}]$ $= \underset{i = 1, \neq k}{\sum^{n}} \frac{a_{i}}{a_{i} - a_{k}} + \lim_{x \to a_{k}^{(+)}} \frac{a_{k}}{a_{k} - x} = - \infty$ $t h a t m e a n s f o r x \in (a_{k}, a_{k + 1}) w i t h k = 1, 2, . . . n - 1$ $f (x) i s s t r i c k l y i n c r e a s i n g$ $\lim_{x \to a_{k}} f (x) = \lim_{x \to a_{k}^{(+)}} f (x) = - \infty$ $\lim_{x \to a_{k + 1}} f (x) = \lim_{x \to a_{k + 1}^{(-)}} f (x) = + \infty$ $i . e . f (x) = c h a s o n e a n d o n l y r o o t i n (a_{k}, a_{k + 1})$ $w i t h c = a n y r e a l n u m b e r, e . g . 2015 .$ $\lim_{x \to - \infty} f (x) = \underset{i = 1}{\sum^{n}} [\lim_{x \to - \infty} \frac{a_{i}}{a_{i} - x}] = + 0$ $t h a t m e a n s f o r x \in (- \infty, a_{1})$ $f (x) i s s t r i c k l y i n c r e a s i n g$ $\lim_{x \to - \infty} f (x) = + 0$ $\lim_{x \to a_{1}} f (x) = \lim_{x \to a_{1}^{(-)}} f (x) = + \infty$ $i . e . w i t h c > 0, e . g . c = 2015$ $f (x) = c h a s o n e a n d o n l y r o o t i n (- \infty, a_{1})$ $\lim_{x \to + \infty} f (x) = \underset{i = 1}{\sum^{n}} [\lim_{x \to + \infty} \frac{a_{i}}{a_{i} - x}] = - 0$ $t h a t m e a n s f o r x \in (a_{n}, + \infty)$ $f (x) i s s t r i c k l y i n c r e a s i n g$ $\lim_{x \to a_{n}} f (x) = \lim_{x \to a_{1}^{(+)}} f (x) = - \infty$ $\lim_{x \to + \infty} f (x) = - 0$ $i . e . w i t h c > 0, e . g . c = 2015$ $f (x) = c h a s n o r o o t i n (a_{n}, + \infty)$ $s u m m a r y :$ $f (x) = c > 0, e . g . c = 2015 h a s$ $o n e a n d o n l y o n e r o o t i n (- \infty, a_{1})$ $o n e a n d o n l y o n e r o o t i n (a_{k}, a_{k + 1}) w i t h k = 1, 2, . . ., n - 1$ $n o r o o t i n (a_{n}, + \infty)$ $t o t a l l y f (x) = c h a s e x a c t l y n r o o t s .$
	Commented byotchereabdullai@gmail.com last updated on 03/Mar/19

	$The great and ideal professor W$
	Commented bymr W last updated on 03/Mar/19

	Commented bymr W last updated on 03/Mar/19

	$w e c a n s e e$ $f (x) = \underset{i = 1}{\sum^{n}} \frac{a_{i}}{a_{i} - x} = 0 h a s e x a c t l y n - 1 r o o t s a n d$ $f (x) = \underset{i = 1}{\sum^{n}} \frac{a_{i}}{a_{i} - x} = c \neq 0 h a s e x a c t l y n r o o t s .$
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