An algebra problem by Karthik Kannan

Algebra Level 3

If the value of

∑ r = 1 ∞ r 3 + ( r 2 + 1 ) 2 ( r 4 + r 2 + 1 ) ( r 2 + r ) \displaystyle\sum_{r=1}^{\infty} \frac{r^{3}+\left( r^{2}+1 \right)^{2}}{(r^{4}+r^{2}+1)(r^{2}+r)}

can be expressed as a b \displaystyle\frac{a}{b} where a a and b b are coprime positive integers, find the value of a + b a+b .


The answer is 5.

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Karthik Kannan by Karthik Kannan , 24, India

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1 solution

Patrick Corn
Apr 12, 2014

After a partial fraction decomposition of the summand, we get ∑ r = 1 ∞ ( 1 r − 1 r + 1 ) + 1 2 ∑ r = 1 ∞ ( 1 r 2 − r + 1 − 1 r 2 + r + 1 ) \sum_{r=1}^{\infty} \left( \frac1{r} - \frac1{r+1} \right) + \frac12 \sum_{r=1}^\infty \left( \frac1{r^2-r+1} - \frac1{r^2+r+1} \right) .

These are both telescoping sums. Both of them are 1. The sum is 1 + 1 2 = 3 2 1 + \frac12 = \frac32 , so the answer is 5 \fbox{5} .

8 Helpful 0 Interesting 0 Brilliant 0 Confused

What is the meaning of telescopic sum

Rohit Singh - 7 years, 2 months ago

yes, I get it.

Mas Mus - 7 years, 2 months ago

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