If 4 + _ p=1 ¹¹ ⁻¹ ( 2^ p-1 1 + 2^ 2p-1 ) = , then is equal to __________.

Step-by-step solution

The general term of the given series is T_p = ⁻¹ ( 2^ p-1 1 + 2^ 2p-1 ) . This can be rewritten by expressing the numerator and denominator in terms of 2^p and 2^ p-1 : T_p = ⁻¹ ( 2^p - 2^ p-1 1 + 2^p 2^ p-1 ) Using the inverse trigonometric identity ⁻¹ ( x - y 1 + xy ) = ⁻¹x - ⁻¹y , we get: T_p = ⁻¹(2^p) - ⁻¹(2^ p-1 ) Summing T_p from p = 1 to 11 gives a telescoping series: _ p=1 ¹¹ T_p = ( ⁻¹(2^1) - ⁻¹(2^0) ) + ( ⁻¹(2^2) - ⁻¹(2^1) ) + + ( ⁻¹(2¹¹) - ⁻¹(2¹⁰) ) Canceling the intermediate terms, we obtain: _ p=1 ¹¹ T