Let displaystyle∑k=1∞ cot -1((k2/8))=((p/q)) π where (p/q) is rational in its lowest form. Find the value of (p+q).

Question

Let displaystyle∑k=1∞ cot -1((k2/8))=((p/q)) π where (p/q) is rational in its lowest form. Find the value of (p+q). (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

T k = tan -1((8/ k 2))= tan -1((2/( k 2)4))= tan -1((2/1+(( k 2)4-1))) = tan -1((2/1+(( k )2+1)(( k )2-1))= tan /-1)(((( k /2)+1)-(( k /2)-1)/1+(( k )2+1)(( k )2-1))

Q. Let k=1∑∞​cot−1(8k2​)=(qp​)π where qp​ is rational in its lowest form. Find the value of (p+q).

Tk​=tan−1(k28​)=tan−1(4k2​2​)=tan−1(1+(4k2​−1)2​) =tan−1(1+(2k​+1)(2k​−1)2​)=tan−1(1+(2k​+1)(2k​−1)(2k​+1)−(2k​−1)​)

Q. Let $k = 1 \sum \infty cot^{- 1} (\frac{k ^{2}}{8}) = (\frac{p}{q}) π$ where $\frac{p}{q}$ is rational in its lowest form. Find the value of $(p + q)$ .