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I can prove the base case and get that $1/3=1/3$, but i can't get any further with the $n+1$ case. Can someone help me?

I was told to conjecture a formula for the sum $$\frac{1}{3} + \frac{1}{{3 \cdot 5}} + \cdots + \frac{1}{{\left( {2n - 1} \right)\left( {2n + 1} \right)}}$$ I thought I figured out that this sum was equal to $$\frac{1}{2} { - \frac{1}{{4n + 2}}} $$ but I'm starting to think I'm wrong about that. After we have the formula, we were told to prove our conjecture using induction.

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    I really can't understand what is the identity the you want to prove. Could you try to typeset it (better if in LaTeX), with all the necessary parentheses, please?2012-09-18
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    Are you sure you have the right expression? For $n = 2$ you get: $\frac{1}{2 \cdot 6} = \frac{1}{2} - \frac{1}{10}$ which is clearly untrue.2012-09-18
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    If $n=1$ you are dividing by $0$. Also please check your parentheses: one extra left on the left side, it is not clear whether $(2n+2)$ should be in the numerator, and you must mean $1/(4n+2)$, not $(1/4)n+2$ or $1/(4n)+2$ on the right. \frac is your friend in this regard.2012-09-18
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    Also, unless there was supposed to be some sum involved, there is no need for induction. Just multiply everything out so as to get rid of fractions ;)2012-09-18
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    Based on your added comment, you want the left to be $\sum_{i=1}^n \frac 1{2i-1}\frac 1{2i+1}=\frac 12 - \frac 1{4n+2}$. Note the $1$'s, not $2$'s and the sum on the left.2012-09-18

4 Answers 4