How to Find Sum of Harmonic Progression

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

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

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

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

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:

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

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

2. Substitute a = 1 and d = -1/n into the formula for the sum of an arithmetic progression:

$$H_n = \sum_{i=1}^n \left(1 + \left(i-1\right)\left(-\frac{1}{n}\right)\right) = n\left(2 - \frac{n-1}{n}\right)$$

3. Simplify the formula:

$$H_n = n\left(\frac{n+1}{n}\right) = n+1$$

Therefore, the sum of the first n terms of a harmonic progression is given by the formula:

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

Example

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

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

Using the formula for the sum of the first n terms of a harmonic progression, we have:

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

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

Sum of Harmonic Progression Formula

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

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

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

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

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

Properties of the Sum of Harmonic Progression 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.
• The sum of the first n terms of a harmonic progression is always less than the natural logarithm of n plus 1.
• The sum of the first n terms of a harmonic progression approaches infinity as n approaches infinity.

Applications of the Sum of Harmonic Progression 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.
• Find 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 a powerful tool that can be used to solve a variety of problems in mathematics and physics. By understanding the properties of this formula, you can use it to your advantage to solve problems that would otherwise be difficult or impossible to solve.

Sum of Infinite Harmonic Progression

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

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

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

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

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

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

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 progression 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$ diverges, then $ \sum\limits_{n=1}^\infty a_n$ also diverges.

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

$$1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} > 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{4} + \dots + \frac{1}{4} = 1 + \frac{1}{2} + \frac{1}{2} + \dots$$

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

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

Solved Examples on Sum of Harmonic Progression

Example 1:

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

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

Solution:

The formula for the sum of the first n terms of a harmonic progression is given by:

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

Substituting n = 10 into the formula, we get:

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

Therefore, the sum of the first 10 terms of the given harmonic progression is 5/11.

Example 2:

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

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

Solution:

Using the formula for the sum of the first n terms of a harmonic progression, we have:

$$H_{20} = \frac{20}{2(20+1)} = \frac{20}{41} = \frac{10}{20.5}$$

Therefore, the sum of the first 20 terms of the given harmonic progression is 10/20.5.

Example 3:

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

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

Solution:

Using the formula for the sum of the first n terms of a harmonic progression, we have:

$$H_{50} = \frac{50}{2(50+1)} = \frac{50}{101} \approx 0.495$$

Therefore, the sum of the first 50 terms of the given harmonic progression is approximately 0.495.

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 progression is:

$$H_n = \frac{n}{2} \left( \frac{1}{a_1} + \frac{1}{a_n} \right)$$

where:

• $H_n$ is the sum of the first $n$ terms of the harmonic progression
• $a_1$ 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, 1/2, 1/3, 1/4, …
• The series 1/2, 1/3, 1/4, 1/5, …
• The series 1/3, 1/4, 1/5, 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 progressions are used to study the motion of objects.
• In mathematics, harmonic progressions are used to study the properties of numbers.

What are some common misconceptions about harmonic progressions?

Some common misconceptions about harmonic progressions include:

• That harmonic progressions are always increasing.
• That harmonic progressions are always decreasing.
• That harmonic progressions are always convergent.

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