What you make of these Mathmatics.

[Pmb]

Where did the last sum come from? Try letting a_n = b_n = 1, m = 2. Then you have 2/(x + 1) = 0 which has no solution. However your sum gives x = -1 which makes the sum undefined.

[CraigD]

Where did the last sum come from? Try letting a_n = b_n = 1, m = 2. Then you have 2/(x + 1) = 0 which has no solution. However your sum gives x = -1 which makes the sum undefined.

You’re right. There needs to be an additional condition attached, to read:

[math]\sum_{n=1}^{m} \frac{1}{a_nx+b_n} \,=\, 0[/math] has solution [math]x=-\frac{\sum_{n=1}^{m} b_n}{\sum_{n=1}^{m}a_n}[/math] when [math]m[/math] is even and [math]\frac{b_i}{a_i} \not = \frac{b_j}{a_j}[/math] for all [math]1 \le a,j \le m[/math], [math]i \not = j[/math].

 

You could say I omitted this for brevity, or out of laziness – the distinction is a fine one. :)