Proving the Convergence of the Series 1/2^2 1/3^2 ... 1/n^2

In this article, we will explore the convergence of the series 1/2^2 1/3^2 ... 1/n^2. We will use mathematical induction and the integral test to prove that the sum of the series approaches a limit as n approaches infinity. Here, we will break down each step in detail.

Using Mathematical Induction

Let's consider the series defined as:

$$S_n sum_{i2}^{n} frac{1}{i^2}$$

Base Case: n 2

For the base case, we consider n2:

$$S_2 frac{1}{2^2} frac{1}{4}$$

The statement is true for n2 since:

$$frac{1}{2} frac{2-1}{2}$$

Inductive Hypothesis

Assume that the statement holds for nk, where:

$$S_k sum_{i2}^{k} frac{1}{i^2} frac{k-1}{k}$$

Inductive Step

We need to show that the statement holds for nk 1:

$$S_{k 1} sum_{i2}^{k 1} frac{1}{i^2}$$

We can express S_{k 1} as:

$$S_{k 1} S_k frac{1}{(k 1)^2} frac{k-1}{k} frac{1}{(k 1)^2}$$

Note that:

$$frac{k-1}{k} frac{1}{(k 1)^2} frac{k(k 1)^2 (k-1)}{k(k 1)^2} frac{k^3 k k - 1}{k(k 1)^2} frac{k^3 2k - 1}{k(k 1)^2}$$

Since:

$$frac{k^3 2k - 1}{k(k 1)^2} frac{k^2 1}{k 1} frac{(k 1) - 1}{k 1} frac{k}{k 1}$$

We have shown that the statement holds for nk 1. Therefore, by the principle of mathematical induction, the given statement is true for all n ≥ 2.

Using the Integral Test

The integral test is a method to determine the convergence of an infinite series. For the series (sum_{i2}^{infty} frac{1}{i^2}), we consider the corresponding improper integral:

$$int_{2}^{infty} frac{1}{x^2} , dx$$

Let's evaluate this integral:

$$int_{2}^{infty} frac{1}{x^2} , dx left[ -frac{1}{x} right]_{2}^{infty} 0 - left(-frac{1}{2}right) frac{1}{2}$$

Since the integral converges to a finite value, by the integral test, the series (sum_{i2}^{infty} frac{1}{i^2}) also converges.

Conclusion

We have successfully proven the convergence of the series (1/2^2 1/3^2 ... 1/n^2) using both mathematical induction and the integral test. The series approaches a limit as n approaches infinity, and the limit is a finite value.