AP EAMCET2005MathematicsSequences and Series
If |a| < 1, b= _ k=1 ^ a^k k , then a is equal to
Options
- A_ k=1 ^ (-1)^k b^k k
- B_ k=1 ^ (-1)^ k-1 b^k k !
- C_ k=1 ^ (-1)^k b^k (k-1) !
- D_ k=1 ^ (-1)^ k-1 b^k (k+1) !
Correct answer
B. _ k=1 ^ (-1)^ k-1 b^k k !
Step-by-step solution
Given that aligned b & = _ k-1 ^ a^k k & = a^1 1 + a^2 2 + a^3 3 + b & =- (1-a) [ |a| < 1] e^ -b & =(1-a) a=1-e^ -b & =1- (1- b 1 ! + b^2 2 ! - b^3 3 ! + ) a & = b 1 ! - b^2 2 ! + b^3 3 ! - a & = _ k=1 ^ (-1)^ k-1 b^k k ! aligned