∑m=1n tan−1⁡2mm4+m2+2=

tan−1⁡2mm4+m2+2=tan−1⁡2m1+m2+m+1m2−m+1=tan−1⁡m2+m+1−tan−1⁡m2−m+1 so that ∑m=1n tan−1⁡2mm4+m2+2=tan1⁡3−tan1⁡1+tan−1⁡7−tan−1⁡3+…+tan−1⁡n2+n+1−tan−1⁡n2−n+1=tan−1⁡n2+n+1−tan−1⁡1=tan−1⁡n2+nn2+n+2