This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here is the solution for question 2b: 2b) Find the sum of the harmonic sequence: 1/12 + 1/24 + 1/36 + 1/48 + 1/60 + Step 1: Identify the type of sequence. The given sequence is (1)/(12), (1)/(24), (1)/(36), (1)/(48), (1)/(60), . Consider the reciprocals of these terms: 12, 24, 36, 48, 60, . This sequence of reciprocals is an arithmetic progression with a first term a_1 = 12 and a common difference d = 24 - 12 = 12. Therefore, the original sequence is a harmonic progression. Step 2: Write the general term of the sequence. The n-th term of the arithmetic progression is A_n = a_1 + (n-1)d = 12 + (n-1)12 = 12 + 12n - 12 = 12n. The n-th term of the harmonic progression is H_n = (1)/(A_n) = (1)/(12n). Step 3: Formulate the sum as an infinite series. The sum of the harmonic sequence is given by the infinite series: S = _n=1^ H_n = _n=1^ (1)/(12n) Step 4: Evaluate the sum. Factor out the constant (1)/(12): S = (1)/(12) _n=1^ (1)/(n) The series _n=1^ (1)/(n) is the harmonic series. The harmonic series is known to diverge, meaning its sum does not approach a finite value. Since the harmonic series diverges, multiplying it by a non-zero constant ((1)/(12)) also results in a divergent series. The sum of the sequence is divergent. The sum diverges That's 2 down. 3 left today — send the next one.