For The Series 1 N 5 Calculate The Error

When working with the series 1/n^5, it's important to understand how to calculate the error associated with partial sums. This guide provides a comprehensive explanation of the process, including formulas, examples, and practical applications.

Understanding the Series

The series in question is the sum of terms where each term is 1 divided by n raised to the fifth power. Mathematically, this can be represented as:

S = Σ (from n=1 to ∞) 1/n^5

This series is a p-series with p = 5, which means it converges because p > 1. The exact sum of this series is known to be π^4/90, but for practical purposes, we often use partial sums and calculate the error.

Calculating the Error

The error when using a partial sum to approximate the infinite series is the difference between the exact sum and the partial sum. For the series 1/n^5, the error can be estimated using the following approach:

Error ≈ Σ (from n=k+1 to ∞) 1/n^5

For large k, this can be approximated by the integral:

Error ≈ ∫ (from k to ∞) 1/x^5 dx = 1/(4k^4)

This approximation works well when k is sufficiently large. The exact error is always less than the first omitted term, which is 1/(k+1)^5, but the integral approximation provides a more precise estimate.

Example Calculation

Let's consider calculating the sum of the first 100 terms of the series and estimating the error:

S₁₀₀ = Σ (from n=1 to 100) 1/n^5 ≈ 1.0369277551433699

Error ≈ 1/(4*100^4) ≈ 2.5 × 10⁻¹⁰

This means that the partial sum S₁₀₀ is accurate to about 10 decimal places. The actual error is even smaller than this estimate suggests.

Practical Applications

Understanding the error in series approximations is crucial in various fields:

Numerical analysis: When approximating integrals or solving differential equations
Physics: In calculations involving infinite series representations of functions
Engineering: When designing algorithms that require precise numerical results
Mathematics education: To teach students about convergence and approximation techniques

By understanding how to calculate and interpret the error in series approximations, professionals can make more accurate calculations and better informed decisions.

Frequently Asked Questions

What is the exact sum of the series 1/n^5?

The exact sum of the infinite series Σ (from n=1 to ∞) 1/n^5 is π^4/90, where π is approximately 3.141592653589793.

How accurate is the integral approximation for the error?

The integral approximation Error ≈ 1/(4k^4) provides a very accurate estimate of the error when k is large. The actual error is always less than this value.

Can I use this method for other p-series?

Yes, the same principles apply to other p-series where p > 1. The error can be approximated using the integral ∫ (from k to ∞) 1/x^p dx = 1/((p-1)k^(p-1)).