Question

prove that 3log 2-1 = 5/1.2.3 + 7/3.4.5 + 9/5.6.7 +....

Answer 1

Answer:

3log 2-1 = 5/1.2.3 + 7/3.4.5 + 9/5.6.7 +....

Step-by-step explanation:

5/1.2.3 + 7/3.4.5 + 9/5.6.7 +....

=> Tₙ  = (2n+3)/(2n-1)(2n)(2n+1)

=>  Tₙ  = 2/(2n-1) -3/2n +1/(2n+1)

=> Tₙ  = 2/(2n-1) -2/2n - 1/2n +1/(2n+1)

=>  Tₙ  = 2(1/(2n-1) -1/2n)  +1/(2n+1)  - 1/2n

∑ Tₙ  = 2 * ∑(1/(2n-1) -1/2n)  + ∑ (1/(2n+1)  - 1/2n )

= 2 * (  1 - 1/2   + 1/3 - 1/4  + ............)  +  ( 1/3 - 1/2 + 1/5 - 1/4 +..................)

log 2 =   1 - 1/2   + 1/3 - 1/4  + ............

log2 - 1 = - 1/2   + 1/3 - 1/4  + ............

=> log2 - 1  = 1/3 - 1/2 + 1/5 - 1/4 +......

= 2 log2 + log2 - 1

= 3log2 - 1

3log 2-1 = 5/1.2.3 + 7/3.4.5 + 9/5.6.7 +....