Prove that : `log_e(n+1)-log_e(n-1)=2[1/n+1/(3n^3)+1/(5n^5)+...]`

Prove that : log_e(n+1)-log_en=2[1/(2n+1)+1/(3(2n+1)^3)+1/(5(2n+1)^5)+...]

Prove that : log_em-log_en=(m-n)/m+1/2((m-n)/m)^2 ..........m,n>0

Prove that : log_ea-log_eb=2[(a-b)/(a+b)+1/3((a-b)/(a+b))^3 +1/5((a-b)/(a+b))^5+...],a > b

prove that :- C_1 - 1/2 C_2 + 1/3 C_3 + ... + (-1)^(n+1) 1/n C_n = 1+1/2 + .... +1/n

If 2x=y^(1/m)+y^(-1/m) , then prove that (x^2-1)y_(n+2)+(2n+1)xy_(n+1)+(n^2-m^2)y_n=0 .

Prove that "^(2n)C_1 + ^(2n)C_3 + .... + ^(2n)C_(2n-1) = 2^(2n-1)

Prove that "^(2n)C_0 + ^(2n)C_2 + .... + ^(2n)C_(2n) = 2^(2n-1)

Evaluate the following: lim_(ntoinfty) ((n^2+n+1)/(5n^2+2n+1))

Show that the curves y = 2^(x) and y = 5^(x) intersect at an angle tan^(-1) |("1n ((5)/(2)))/(1 + 1n 2 1n 5)| . Note Angle between two curves is the angle between their tangents at the point of intersection.