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Convergent and Divergent Series Calculator

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Example
Created on 2024-06-20Asked by Harper Brown (Solvelet student)
Determine if the series ∑n=1∞(−1)nn \sum_{n=1}^{\infty} \frac{(-1)^n}{n} converges absolutely, conditionally, or diverges.

Solution

Step 1: Identify the series ∑n=1∞(−1)nn \sum_{n=1}^{\infty} \frac{(-1)^n}{n} . Step 2: Test for absolute convergence: ∑n=1∞∣(−1)nn∣=∑n=1∞1n. \sum_{n=1}^{\infty} \left| \frac{(-1)^n}{n} \right| = \sum_{n=1}^{\infty} \frac{1}{n}. Step 3: The series ∑n=1∞1n \sum_{n=1}^{\infty} \frac{1}{n} is the harmonic series, which diverges. Step 4: Test for conditional convergence: Since the series ∑n=1∞1n \sum_{n=1}^{\infty} \frac{1}{n} diverges, we need to test the series ∑n=1∞(−1)nn \sum_{n=1}^{\infty} \frac{(-1)^n}{n} for convergence. Step 5: Apply the alternating series test: The series ∑n=1∞(−1)nn \sum_{n=1}^{\infty} \frac{(-1)^n}{n} is alternating and lim⁡n→∞1n=0 \lim_{n \to \infty} \frac{1}{n} = 0 , so it converges conditionally. Step 6: Conclusion. The series ∑n=1∞(−1)nn \sum_{n=1}^{\infty} \frac{(-1)^n}{n} converges conditionally. Solved on Solvelet with Basic AI Model
Some of the related questions asked by Michael White on Solvelet
1. Determine whether the series ∑n=1∞1n2 \sum_{n=1}^{\infty} \frac{1}{n^2} is convergent or divergent2. Find the sum of the convergent geometric series ∑n=0∞(12)n \sum_{n=0}^{\infty} \left( \frac{1}{2} \right)^n .,
DefinitionA series in mathematics (books/logs) is defined to be the sum of a sequence of terms. A series whose sequence of partial sums approaches a finite limit as the count of terms is increased is said to be a convergent series. On the other side, a series that does not have its partial sum sequence converge to a definite limit is a divergent series. In mathematics, statements are essential concepts in calculus, real analysis, and mathematical analysis for studying infinite sums and their convergence properties. The convergence of series is important for evaluating mathematical expressions and solving many problems in physics, engineering, and other fields of scientific disciplines.
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