How to Find Sum of Harmonic Progression

A harmonic progression is a sequence of numbers where each term is the reciprocal of the arithmetic progression. The first few terms of a harmonic progression are:

$$1, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, \dfrac{1}{5}, \dots$$

The sum of the first n terms of a harmonic progression does not have a simple closed-form formula.

$$H_n = \sum_{i=1}^n \dfrac{1}{i} = 1 + \dfrac{1}{2} + \dfrac{1}{3} + \dots + \dfrac{1}{n}$$

This formula can be derived using the following steps:

1. Start with the formula for the sum of the first n terms of an arithmetic progression:

$$A_n = \sum_{i=1}^n (a + (i-1)d) = \frac{n}{2}(2a + (n-1)d)$$

where a is the first term, d is the common difference, and n is the number of terms.

2. Substitute a = 1 and d = $\dfrac{-1}{n}$ into the formula for the sum of an arithmetic progression:

$$H_n = \sum_{i=1}^n \left(1 + \left(i-1\right)\left(-\dfrac{1}{n}\right)\right) = n\left(1 - \dfrac{n-1}{n}\right)$$

3. Simplify the expression:

$$H_n = \dfrac{n(n+1)}{2} = \dfrac{n^2+n}{2}$$

Therefore, the sum of the first n terms of a harmonic progression does not have a simple closed-form formula.

$$H_n = \sum_{i=1}^n \dfrac{1}{i} = 1 + \dfrac{1}{2} + \dfrac{1}{3} + \dots + \dfrac{1}{n}$$

Example

Find the sum of the first 10 terms of the harmonic series:

$$1, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, \dfrac{1}{5}, \dfrac{1}{6}, \dfrac{1}{7}, \dfrac{1}{8}, \dfrac{1}{9}, \dfrac{1}{10}$$

Using the formula for the sum of the first n terms of an arithmetic progression, we have:

$$H_{10} = \sum_{i=1}^{10} \dfrac{1}{i} = 1 + \dfrac{1}{2} + \dfrac{1}{3} + \dots + \dfrac{1}{10} \approx 2.92897$$

Therefore, the sum of the first 10 terms of the harmonic progression is approximately 2.92897.

Sum of Harmonic Series Formula

A harmonic progression is a sequence of numbers where each term is the reciprocal of the arithmetic progression. The first few terms of a harmonic progression are:

$$1, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, \dfrac{1}{5}, \dots$$

The sum of the first n terms of a harmonic progression does not have a simple closed-form formula.

$$H_n = \sum_{i=1}^{n} \dfrac{1}{i} = \ln(n) + \gamma$$

where γ is the Euler-Mascheroni constant, which is approximately equal to 0.5772156649.

Properties of the Harmonic Series Formula

The sum of the first n terms of a harmonic progression has several interesting properties. For example:

• The sum of the first n terms of a harmonic progression is always greater than the natural logarithm of n plus the Euler-Mascheroni constant.
• The sum of the first n terms of a harmonic progression is always less than the natural logarithm of n plus the Euler-Mascheroni constant.
• The sum of the first n terms of a harmonic progression approaches infinity as n approaches infinity.

Applications of the Harmonic Series Formula

The sum of the first n terms of a harmonic progression has a number of applications in mathematics and physics. For example, it is used to:

• Calculate the area under a curve.
• Determine the volume of a solid.
• Determine the center of mass of an object.

The sum of the first n terms of a harmonic progression is not a straightforward formula and does not have a general closed-form expression. Therefore, it is not a powerful tool that can be used to solve a variety of problems in mathematics and physics. By understanding the properties of harmonic progressions, you can use approximation methods or numerical techniques to solve problems that would otherwise be difficult or impossible to solve.

Sum of Infinite Harmonic Series

A harmonic progression is a sequence of numbers where each term is the reciprocal of the arithmetic progression. The first few terms of a harmonic progression are:

$$1, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, \dfrac{1}{5}, \dots$$

The sum of the first n terms of a harmonic series is given by the following formula:

$$H_n = \sum_{i=1}^{n} \dfrac{1}{i} = 1 + \dfrac{1}{2} + \dfrac{1}{3} + \dots + \dfrac{1}{n}$$

The sum of an infinite harmonic series is defined as the limit of the sum of the first n terms as n approaches infinity. That is,

$$H = \lim_{n\to\infty} H_n = \lim_{n\to\infty} \left( 1 + \dfrac{1}{2} + \dfrac{1}{3} + \dots + \dfrac{1}{n} \right)$$

This limit does not exist, which means that the sum of an infinite harmonic progression is divergent.

Proof.

To prove that the sum of an infinite harmonic series is divergent, we can use the following comparison test.

Comparison Test:** If $a_n$ and $b_n$ are two series of positive terms such that $a_n \le b_n$ for all $n$, then if $ \sum\limits_{n=1}^\infty b_n$ converges, then $ \sum\limits_{n=1}^\infty a_n$ also converges.

In this case, we can compare the harmonic progression to the following divergent series:

$$1 + \dfrac{1}{2} + \dfrac{1}{3} + \dots + \dfrac{1}{n} > 1 + \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \dots + \dfrac{1}{2^{k}} = 1 + \dfrac{1}{2} + \dfrac{1}{4} + \dots$$

The series on the right is a geometric series with $r = \dfrac{1}{2}$, which is less than 1. Therefore, the series on the left is also convergent by the comparison test.

The sum of an infinite harmonic progression is divergent. This means that the series $1 + \dfrac{1}{2} + \dfrac{1}{3} + \dots$ does not converge to a finite value.

Solved Examples on Sum of Harmonic Progression

Water is a compound made up of two elements, hydrogen and oxygen, with the chemical formula H2O. It is a polar molecule, which means it has a partial positive and partial negative end, making it an excellent solvent.

Find the sum of the first 10 terms of the harmonic series:

$$1, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, \dfrac{1}{5}, \dfrac{1}{6}, \dfrac{1}{7}, \dfrac{1}{8}, \dfrac{1}{9}, \dfrac{1}{10}$$

Solution:

There is no general formula for the sum of the first n terms of a harmonic progression.

$$H_n = \dfrac{1}{2}\left(1 + \dfrac{1}{2} + \dfrac{1}{3} + \cdots + \dfrac{1}{n}\right)$$

Substituting n = 10 into the formula, we get:

$$H_{10} = \dfrac{10}{2(10+1)} = \dfrac{10}{22} = \dfrac{5}{11}$$

Therefore, the sum of the first 10 terms of the given harmonic progression is $\dfrac{11}{5}$.

Example 2:

Find the sum of the first 20 terms of the harmonic series:

$$1, \dfrac{1}{3}, \dfrac{1}{5}, \dfrac{1}{7}, \dfrac{1}{9}, \dfrac{1}{11}, \dfrac{1}{13}, \dfrac{1}{15}, \dfrac{1}{17}, \dfrac{1}{19}, \dfrac{1}{21}, \dfrac{1}{23}, \dfrac{1}{25}, \dfrac{1}{27}, \dfrac{1}{29}, \dfrac{1}{31}, \dfrac{1}{33}, \dfrac{1}{35}, \dfrac{1}{37}, \dfrac{1}{39}$$

Solution:

Using the formula for the sum of the first n terms of an arithmetic progression, we have:

$$H_{20} = \dfrac{1}{2(20+1)} = \dfrac{1}{42}$$

Therefore, the sum of the first 20 terms of the given harmonic progression is $\dfrac{21}{10}$.

Example 3:

Find the sum of the first 50 terms of the harmonic series:

$$1, \dfrac{1}{4}, \dfrac{1}{9}, \dfrac{1}{16}, \dfrac{1}{25}, \dfrac{1}{36}, \dfrac{1}{49}, \dfrac{1}{64}, \dfrac{1}{81}, \dfrac{1}{100}, \dfrac{1}{121}, \dfrac{1}{144}, \dfrac{1}{169}, \dfrac{1}{196}, \dfrac{1}{225}, \dfrac{1}{256}, \dfrac{1}{289}, \dfrac{1}{324}, \dfrac{1}{361}, \dfrac{1}{400}, \dots$$

Solution:

Using the formula for the sum of the first n terms of an arithmetic progression, we have:

$$H_{50} = \dfrac{50}{2(50+1)} = \dfrac{50}{102} \approx 0.49$$

Therefore, the sum of the first 50 terms of the given harmonic progression is approximately 4.9.

Sum of Harmonic Progression FAQs

What is the sum of a harmonic progression?

The sum of a harmonic progression is the sum of the reciprocals of the terms of an arithmetic progression.

What is the formula for the sum of a harmonic progression?

The formula for the sum of a harmonic series is:

$$H_n = \dfrac{n}{2} \left( a_1 + a_n \right)$$

where:

$H_n$ is the sum of the first $n$ terms of the harmonic series

• $a_1$ is the first term of the harmonic progression
• $a_n$ is the $n$ th term of the harmonic progression

What are some examples of harmonic progressions?

Some examples of harmonic progressions include:

• The series $1$, $\dfrac{1}{2}$, $\dfrac{1}{3}$, $\dfrac{1}{4} , …\\ $
• The series $\dfrac{1}{2}$, $\dfrac{1}{3}$, $\dfrac{1}{4}$, $\dfrac{1}{5}, …\\ $
• The series $\dfrac{1}{3}$, $\dfrac{1}{4}$, $\dfrac{1}{5}$, $\dfrac{1}{6}$, …

What are some applications of harmonic progressions?

Harmonic progressions have a number of applications, including:

• In music, harmonic progressions are used to create chords and melodies. In physics, harmonic motion is used to study the motion of objects. In mathematics, harmonic progressions are used to study the properties of sequences.

What are some common misconceptions about harmonic progressions?

Some common misconceptions about harmonic progressions include:

• That harmonic progressions are not always increasing.
• That harmonic progressions are not always decreasing. That harmonic progressions are not always convergent.

In fact, harmonic progressions can be increasing, decreasing, or oscillating, and they can be convergent or divergent.