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COMEDK2021MathematicsSequences and Series

The value of 1 1 !+2 2 !+3 3 !+ +n n ! is

Options

  1. A(n+1) !
  2. B(n+1) !+1
  3. C(n+1) !-1
  4. DNone of these

Correct answer

C. (n+1) !-1

Step-by-step solution

The general term of the series is T_r = r r! . We can rewrite the general term as T_r = (r+1-1) r! = (r+1) r! - r! . Since (r+1) r! = (r+1)! , we have T_r = (r+1)! - r! . The sum S_n = _ r=1 ^ n T_r = _ r=1 ^ n ((r+1)! - r!) . This is a telescoping series: S_n = (2! - 1!) + (3! - 2!) + (4! - 3!) + + ((n+1)! - n!) . All intermediate terms cancel out, leaving S_n = (n+1)! - 1! . Therefore, S_n = (n+1)! - 1 . Answer: (n+1) !-1

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