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Convergence and Divergence Calculator

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Example
Created on 2024-06-20Asked by Jacob Thomas (Solvelet student)
Determine if the series ∑n=1∞1n2 \sum_{n=1}^{\infty} \frac{1}{n^2} converges or diverges.

Solution

Step 1: Identify the series ∑n=1∞1n2 \sum_{n=1}^{\infty} \frac{1}{n^2} . Step 2: Use the p-series test: ∑n=1∞1np \sum_{n=1}^{\infty} \frac{1}{n^p} converges if p>1 p > 1 and diverges if p≤1 p \leq 1 . Step 3: Since p=2>1 p = 2 > 1 , the series ∑n=1∞1n2 \sum_{n=1}^{\infty} \frac{1}{n^2} converges. Step 4: Conclusion. The series ∑n=1∞1n2 \sum_{n=1}^{\infty} \frac{1}{n^2} converges. Solved on Solvelet with Basic AI Model
Some of the related questions asked by Amelia Rodriguez on Solvelet
1. Determine whether the series ∑n=1∞1n \sum_{n=1}^{\infty} \frac{1}{n} converges or diverges2. Apply the alternating series test to determine the convergence of the series ∑n=1∞(−1)n+1n \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} .,
DefinitionConvergence and Divergence are the properties of sequences and series which tell about how these sequence and series behave if no of terms increases. A convergent sequence is one whose terms get closer and closer to a finite limit as the sequence progresses, and a convergent series is one whose sum of its terms approaches a finite value. In mathematics, divergence in a sequence or a series is such that: In the disciplines of mathematical analysis, calculus, and real analysis, convergence and divergence are common terms used in elevating functions and sequences in proximity and in quality. For instance, the sequence an​​=n1​ →0 as n→∞, but the harmonic series ∑n=1∞​n1​ is divergent.
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