Proving the Summation of Square Roots Using Telescoping Sums

In this article, we will explore the solution to a specific summation problem involving square roots. The goal is to find an accurate approximation of the sum and validate its correctness using a telescoping sum and Riemann sums.

Problem Statement

Given the sum:

Sigma;k1 to 2025 1/sqrt{k}

We aim to find a method to prove and approximate this sum.

Using a Telescoping Sum

We start by attempting to find a telescoping sum for the terms. A telescoping sum is a sum in which most terms cancel out:

1/sqrt{k} 2sqrt{k 1} - 2sqrt{k}

For n 2025, this gives the sum to n of 2sqrt{n 1} - 2sqrt{1}. Plugging in n 2025, we get:

2sqrt{2026} - 2sqrt{1}

This is not a precise approximation, so we refine our approach.

Refined Approach Using Inequalities

We utilize the inequality:

1/sqrt{k} 2/(sqrt{k} times; sqrt{k-1})

We can then bound the sum of the last 2024 terms by a telescoping sum:

1/2 1/3 ... 1/2025 2/(2 times; 1) 2/(3 times; 2) ... 2/(2025 times; 2024)

Simplifying the telescoping sum:

2sqrt{2} - sqrt{1} 2sqrt{3} - sqrt{2} ... 2sqrt{2025} - sqrt{2024}

This simplifies to:

2sqrt{2025} - sqrt{1} 245 - 1 88

Motivating the Solution Using Riemann Sums

To motivate the solution, we use the concept of Riemann sums. We consider the sum:

Sigma;k1 to 4 1/sqrt{k}

This can be thought of as a right endpoint Riemann sum, which approximates the area under the curve y 1/sqrt{x}. The sum of the areas of four rectangles lies below the curve:

Sigma;k2 to 4 1/sqrt{k} int;1 to 4 1/x dx

Adding 1 to both sides gives:

1 Sigma;k1 to 4 1/sqrt{k} 1 int;1 to 4 1/x dx

Evaluating the integral:

1 int;1 to 4 1/x dx 1 2(sqrt{4} - sqrt{1}) 1 2(2 - 1) 3

Generalizing to a Larger Interval

For the interval [1, 2025], we use the same idea:

Sigma;k1 to 2025 1/sqrt{k} 1 int;1 to 2025 1/x dx

Evaluating the integral:

1 2(sqrt{2025} - sqrt{1}) 1 2(45 - 1) 1 88 89

Therefore, we conclude that:

Sigma;k1 to 2025 1/sqrt{k} 89

Conclusion

This approach provides a rigorous method to approximate and prove the sum of square roots using both telescoping sums and Riemann sums. The final result shows that the sum is bounded above by 89.

/p

pFor further reading and more advanced topics on summations and approximations, explore the following sections:

liMore on Telescoping Sums/li liInequalities in Calculus/li liRiemann Sums and Numerical Integration/li /ul

/p