This English question involves literary analysis, grammar, or writing skills. The detailed response below provides a well-structured answer with supporting evidence and clear explanations.
ScanSolve AI Answer
The image is very blurry, but it appears to be asking for the sum of an infinite series. Based on the visible symbols, I will assume the question is to evaluate the sum of the series _n=1^ (1)/(n(n+1)). Step 1: Decompose the general term using partial fractions. The general term of the series is a_n = (1)/(n(n+1)). We can express this term as a difference of two fractions: (1)/(n(n+1)) = (A)/(n) + (B)/(n+1) To find A and B, we multiply both sides by n(n+1): 1 = A(n+1) + Bn Set n=0: 1 = A(0+1) + B(0) 1 = A. Set n=-1: 1 = A(-1+1) + B(-1) 1 = -B B = -1. So, the general term can be written as: (1)/(n(n+1)) = (1)/(n) - (1)/(n+1) Step 2: Write out the partial sum S_N. The partial sum S_N is the sum of the first N terms: S_N = _n=1^N ((1)/(n) - (1)/(n+1)) Let's write out the first few terms and the last term: S_N = (1 - (1)/(2)) + ((1)/(2) - (1)/(3)) + ((1)/(3) - (1)/(4)) + + ((1)/(N) - (1)/(N+1)) Step 3: Identify the telescoping sum. Notice that most of the terms cancel each other out. This is a telescoping series: S_N = 1 - (1)/(2) + (1)/(2) - (1)/(3) + (1)/(3) - (1)/(4) + + (1)/(N) - (1)/(N+1) The partial sum simplifies to: S_N = 1 - (1)/(N+1) Step 4: Find the sum of the infinite series. To find the sum of the infinite series, we take the limit of the partial sum as N approaches infinity: _n=1^ (1)/(n(n+1)) = _N S_N = _N (1 - (1)/(N+1)) As N , the term (1)/(N+1) approaches 0. _N (1 - (1)/(N+1)) = 1 - 0 = 1 The sum of the series is: 1 Drop the next question.