Determine the convergence or divergence of infinite series instantly. Whether you are dealing with Geometric Series or p-Series, this tool provides precise calculations and step-by-step mathematical proofs.
Converge or Diverge Calculator
Converge or Diverge Calculator Formula:
p-Series: p > 1 (Converges)
Formula Source: Wolfram MathWorld – Convergence Tests
Variables:
- a (Initial Term): The first term of the series.
- r (Common Ratio): The factor by which each term is multiplied to get the next.
- p (Power): The exponent of ‘n’ in the denominator of a p-series.
Related Calculators:
- Limit Comparison Test Calculator
- Ratio Test Calculator
- Infinite Series Sum Calculator
- Integral Test Solver
What is Converge or Diverge Calculator?
A Converge or Diverge Calculator is a specialized mathematical tool used to determine if an infinite series approaches a finite value (converges) or grows without bound (diverges). In calculus, series analysis is crucial for approximating functions and understanding physical systems.
This calculator specifically automates the Geometric Series Test and the p-Series Test, two of the most fundamental criteria used in mathematical analysis.
How to Calculate (Example):
- Identify the type of series (e.g., Geometric).
- Determine the common ratio r. If the series is 1 + 1/2 + 1/4…, then r = 0.5.
- Check the condition: Since |0.5| < 1, the series converges.
- For p-series (1/n²), identify p = 2. Since 2 > 1, it converges.
Frequently Asked Questions (FAQ):
What is the difference between convergence and divergence? Convergence means the sum of the series approaches a specific number, while divergence means the sum goes to infinity or fluctuates forever.
Does the harmonic series converge? No, the harmonic series (1/n) is a p-series where p=1. Since p is not greater than 1, it diverges.
What if |r| is exactly 1? In a geometric series, if |r| = 1, the series diverges because the terms do not approach zero.
Can a series converge if the limit of terms is not zero? No, according to the Divergence Test, if the limit of the terms is not zero, the series must diverge.