The value of 12 displaystyle ∑ r = 1∞ ((12 r2 + 1)/(64 r6 - 48 r4 + 12 r2 - 1)) is :

Question

The value of 12 displaystyle ∑ r = 1∞ ((12 r2 + 1)/(64 r6 - 48 r4 + 12 r2 - 1)) is : (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

6 displaystyle ∑ r = 110(24 r2 + 2/ 64 r6 - 48 r4 + 12 r2 - 1) =6 displaystyle ∑ r = 1∞ (24 r2 + 2/(4 r2 - 1)3)=6 displaystyle ∑ Tr ⇒ Tr=(24 r2 + 2/(2 r - 1)3 (2 r + 1)3) =(1/(2 r - 1)3)-(1/(2 r + 1)3) displaystyle ∑ r = 1∞ Tr=1

Q. The value of 12r=1∑∞​(64r6−48r4+12r2−1)(12r2+1)​ is :

6r=1∑10​64r6−48r4+12r2−124r2+2​ =6r=1∑∞​(4r2−1)324r2+2​=6∑Tr​ ⇒Tr​=(2r−1)3(2r+1)324r2+2​ =(2r−1)31​−(2r+1)31​ r=1∑∞​Tr​=1

Q. The value of $12 r = 1 \sum \infty \frac{( 12 r ^{2} + 1 )}{( 64 r ^{6} - 48 r ^{4} + 12 r ^{2} - 1 )}$ is :