Introduction

If you are faced with the task of finding the sum of the series 1/2 × 2/3 × 3/4 × × 49/50, this article is for you. Through a series of innovative and practical steps, we will break down this complex problem, providing a detailed guide that is both easy to understand and follow.

Understanding the Series

The given series is:

S 1/2 × 2/3 × 3/4 × 49/50

To simplify this, we can break down each term into a more manageable form. This approach not only unwraps the complexity of the series but also reveals underlying mathematical structures that can be utilized for further simplification and analysis.

Expressing Each Term

Consider the nth term of the series:

1/n × (n/(n 1)) 1 - 1/(n 1)

By applying this transformation to each term, the series can be rewritten as:

S (1 - 1/2) (1 - 1/3) (1 - 1/4) (1 - 1/50)

This transformation allows us to express the series as a sum of simpler terms, significantly simplifying our task.

Separating the Series

Now, let's separate the series into two parts:

S (1 1 1 1) - (1/2 1/3 1/4 1/50)

The first part is straightforward:

1 1 1 1 49

The second part is a harmonic series, which is the sum of the reciprocals of the first 50 natural numbers. This can be simplified as follows:

1/2 1/3 1/4 1/50 H_{50} - 1

Where H_{50} is the 50th harmonic number.

Summing the Series

Putting it all together, we get:

S 49 - (H_{50} - 1) 50 - H_{50}

Thus, the sum of the series S is:

S 50 - H_{50}

Approximating the Harmonic Number

To find the numeric value of H_{50}, we can use the approximation:

H_n ≈ ln(n) γ

Where γ is the Euler-Mascheroni constant, approximately equal to 0.577.

Calculating H_{50}:

H_{50} ≈ ln(50) 0.577 ≈ 3.912 0.577 ≈ 4.489

Thus:

S ≈ 50 - 4.489 ≈ 45.511

Conclusion

In conclusion, the sum of the series 1/2 × 2/3 × 3/4 × × 49/50 can be expressed as 50 - H_{50}, where H_{50} is the 50th harmonic number. Using the approximation for harmonic numbers, we find that the sum is approximately 45.511. This method not only solves the problem but also provides insights into the properties and behavior of harmonic numbers.