Series and sequences

**AuXian** · October 4th 2012, 02:41 AM

Can somebody teach me how to solve this question? Please…

Sorry... since i don't know how to write the symbols here... so i used attachment...

**emakarov** · October 4th 2012, 03:06 AM

Originally Posted by AuXian

Show that $\sum_{n=1}^r\frac{r^2+r^{-1}}{r^2+r}=\frac{n^2}{n+1}$

So, are you summing r constants (i.e., expressions that don't depend on n)?

**johnsomeone** · October 4th 2012, 03:25 AM

If it's $\sum_{r = 1}^n \frac{r^2 + r^{-1}}{r^2 + r} = \frac{n^2}{n+1}$ , then it isn't even true.

$n =1:$ LHS $= \frac{1^2 +1^{-1}}{1^2 + 1} = \frac{2}{2} = 1$ . RHS $= \frac{1^2}{1+1} = \frac{1}{2}$ .

$n =2$ : LHS $= (1) + \frac{2^2 +2^{-1}}{2^2 + 2} = 1 + \frac{4.5}{6} = \frac{7}{4}$ . RHS $= \frac{2^2}{2+1} = \frac{4}{3}$ .

**AuXian** · October 4th 2012, 06:13 AM

Originally Posted by emakarov

So, are you summing r constants (i.e., expressions that don't depend on n)?

ya... only the final step will convert r to n... It is needed to show that LHS=RHS, the LHS will be used. My problem is, I don't know how to simplify the earlier part (about r)...

**emakarov** · October 4th 2012, 07:42 AM

Originally Posted by AuXian

only the final step will convert r to n...

I am not sure what this means. Please write the question correctly.

**AuXian** · October 4th 2012, 08:09 AM

Originally Posted by emakarov

I am not sure what this means. Please write the question correctly.

What i wrote there is a complete question....

**emakarov** · October 4th 2012, 08:15 AM

Originally Posted by AuXian

What i wrote there is a complete question....

And yet you had to add the phrase "only the final step will convert r to n," the meaning of which I don't understand. Even worse, the left-hand side of the equation in the quote in post #2 does not depend on n (n is a bound variable used as the summation index) while the right hand side does. And if n and r are reversed, then, as post #3 shows, the equation is incorrect.

**cobooboc** · October 4th 2012, 07:34 PM

Originally Posted by johnsomeone

If it's $\sum_{r = 1}^n \frac{r^2 + r^{-1}}{r^2 + r} = \frac{n^2}{n+1}$ , then it isn't even true.

$n =1:$ LHS $= \frac{1^2 +1^{-1}}{1^2 + 1} = \frac{2}{2} = 1$ . RHS $= \frac{1^2}{1+1} = \frac{1}{2}$ .

$n =2$ : LHS $= (1) + \frac{2^2 +2^{-1}}{2^2 + 2} = 1 + \frac{4.5}{6} = \frac{7}{4}$ . RHS $= \frac{2^2}{2+1} = \frac{4}{3}$ .

I'm always very envious of you, you can always write the formula well.

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