Problem 92
Question
If \(H_{n}=1+\frac{1}{2}+\frac{1}{3}+\ldots . .+\frac{1}{n}\), then the value of \(1+\frac{3}{2}+\frac{5}{3}+\ldots .+\frac{2 n-1}{n}\) is (A) \(n-H_{n}\) (B) \(2 n-H_{n}\) (C) \((n-1)-H_{n}\) (D) \(n-2 H_{n}\)
Step-by-Step Solution
Verified Answer
(B) \(2n - H_n\)
1Step 1: Analyze the given sequence
We are tasked with finding the value of the sequence: \(1 + \frac{3}{2} + \frac{5}{3} + \cdots + \frac{2n-1}{n}\). Notice the fractions in the sequence have the form \(\frac{2k-1}{k}\) for \(k = 1\) to \(n\).
2Step 2: Rewrite the sequence as summation
Express the sequence using summation: \[ \sum_{k=1}^{n} \frac{2k-1}{k} \]. This can be split into two summations: \[ \sum_{k=1}^{n} \frac{2k-1}{k} = \sum_{k=1}^{n} 2 - \sum_{k=1}^{n} \frac{1}{k} \].
3Step 3: Simplify using known series
Recognize that \(\sum_{k=1}^{n} \frac{1}{k} = H_n\) (the nth harmonic number). Thus, the expression becomes \[ \sum_{k=1}^{n} 2k/k - \sum_{k=1}^{n} \frac{1}{k} = 2n - H_n \].
4Step 4: Verify against options
The simplified expression \(2n - H_n\) matches option (B).
Key Concepts
Sequence SummationMathematical SeriesHarmonic Series
Sequence Summation
A sequence summation is an operation where we add up the elements of a sequence. Think about a sequence as a list of numbers where order matters. For example, in the sequence \(1, \frac{3}{2}, \frac{5}{3}, \ldots , \frac{2n-1}{n}\), we want to find the total sum from the first to the last term. The notation \(\sum\) is often used because it represents the sum over a sequence or series. In this case, the sequence is expressed in a general form \(\frac{2k-1}{k}\) with \(k\) running from 1 to \(n\).
Here’s the beauty of sequence summation: it simplifies the often complex-looking individual terms into a single, manageable expression. By arranging and combining terms appropriately, one can uncover hidden patterns or relationships within the sequence. This allows for more straightforward computations and better understanding of the behavior of the sequence.
Here’s the beauty of sequence summation: it simplifies the often complex-looking individual terms into a single, manageable expression. By arranging and combining terms appropriately, one can uncover hidden patterns or relationships within the sequence. This allows for more straightforward computations and better understanding of the behavior of the sequence.
Mathematical Series
A mathematical series is the sum of the terms of a sequence. It's like collecting a series of steps in a process where each adds more to the total result. The term 'series' usually implies that we are considering the sum rather than the ordered list itself. In a formula, it is often presented using the summation notation: \(\sum_{k=1}^{n} a_k\), where \(a_k\) are the terms of the sequence.
For our specific problem, the sequence is composed of terms \(\frac{2k-1}{k}\). When these are summed from \(k=1\) to \(n\), it forms a series. By breaking down the sum into two separate parts, \(\sum_{k=1}^{n} 2 - \sum_{k=1}^{n} \frac{1}{k}\), we can identify the formulae involved as simple and useful elements. This is a common practice in evaluating mathematical series to simplify expressions and make calculations easier.
For our specific problem, the sequence is composed of terms \(\frac{2k-1}{k}\). When these are summed from \(k=1\) to \(n\), it forms a series. By breaking down the sum into two separate parts, \(\sum_{k=1}^{n} 2 - \sum_{k=1}^{n} \frac{1}{k}\), we can identify the formulae involved as simple and useful elements. This is a common practice in evaluating mathematical series to simplify expressions and make calculations easier.
Harmonic Series
The Harmonic Series is a specific type of mathematical series where each term is the reciprocal of an integer: \(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}\). The sum of the first \(n\) terms of a harmonic series is referred to as the \(n\)-th harmonic number, denoted \(H_n\).
Understanding harmonic numbers and series is crucial because they appear frequently in mathematical problems and theories. They help us unravel the patterns within number theory and are key components when analyzing the sum of reciprocals. In the context of our problem, \(\sum_{k=1}^{n} \frac{1}{k}\) simplifies to \(H_n\), which helps us identify part of the sequence we need to subtract in our sum simplification. Hence, knowing the properties and behavior of harmonic series allows us to proceed efficiently through complex calculations.
Understanding harmonic numbers and series is crucial because they appear frequently in mathematical problems and theories. They help us unravel the patterns within number theory and are key components when analyzing the sum of reciprocals. In the context of our problem, \(\sum_{k=1}^{n} \frac{1}{k}\) simplifies to \(H_n\), which helps us identify part of the sequence we need to subtract in our sum simplification. Hence, knowing the properties and behavior of harmonic series allows us to proceed efficiently through complex calculations.
Other exercises in this chapter
Problem 90
Let there be \(n\) numbers in G.P. whose common ratio is \(r\) and \(S_{m}\) denotes the sum of their first \(m\) terms. The sum of their products taken two at
View solution Problem 91
If \(a, b, c, d\) are distinct integers in A.P. such that \(d=a^{2}\) \(+b^{2}+c^{2}\), then \(a+b+c+d=\) (A) 2 (B) 1 (C) 0 (D) None of these
View solution Problem 93
If \(a_{m}\) be the \(m\) th term of an A.P., then \(a_{1}^{2}-a_{2}^{2}+a_{3}^{2}-a_{4}^{2}+\ldots .+a_{2 n-1}^{2}-a_{2 n}^{2}=\) (A) \(\frac{n-1}{2 n-1}\left(
View solution Problem 94
If \(a_{n+1}=\frac{1}{1-a_{n}}\) for \(n \geq 1\) and \(a_{3}=a_{1}\), then \(\left(a_{2001}\right)^{2001}=\) (A) \(\mathbb{1}\) (B) \(-1\) (C) 0 (D) None of th
View solution