Sequences & Series
Telescoping sum via partial fractions
Grade Class 12
Question:
The value of $\displaystyle\sum_{r=1}^{\infty}\frac{8r}{4r^4+1}$ is
Step-by-Step Solution
Key Concept: $4r^4+1=(2r^2-2r+1)(2r^2+2r+1)$. $8r=2[(2r^2+2r+1)-(2r^2-2r+1)]$. So each term $=2(1/(2r^2-2r+1)-1/(2r^2+2r+1))$. Telescoping: sum $=2/1=2$.
Sum $=2$.
Correct Answer: 2