Derive an expression for the sum of the reciprocals of two consecutive odd integers 2m-1 and 2m+1

Question

Derive an expression for the sum of the reciprocals of two consecutive odd integers 2m-1 and 2m+1

Answer 1

2m - 1 = an odd integer

2m + 1 = the next odd integer

 

Their reciprocals are, respectively, 1/(2m - 1) and 1/(2m + 1).

 

The sum of their reciprocals is

 

sum = 1/(2m - 1) + 1/(2m + 1)

 

If all you need is an expression for the sum, then this is the answer.  You can simplify it further if you wish.

 

 

Answer 2

1/(2m-1) + 1/(2m+1) = [(2m+1) + (2m-1)]/(4m²-1)

 

                                     = 4m/(4m²-1)