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Deviation statistics: difference between the realization of a variable and some value (e.g., mean). Statistics of the deviation distributions consist of standard deviation, average absolute deviation, median absolute deviation , maximum absolute deviation.
Sometimes proving that a series diverges can be quite a challenge! Using the Divergence Test, also called the \(n^{th}\) Term Test for Divergence, is a simple test you can do to see right away if a series diverges. This can save you considerable time in the long run.
Nov 21, 2023 · Convergence and Divergence Tests: Examples. Lesson Summary. Frequently Asked Questions. How do you test for convergence and divergence? Some tests will determine if a series is convergent or...
- Some tests will determine if a series is convergent or divergent- these tests include the limit comparison test and the comparison test. The compar...
- Divergence testing only tests for divergence, while convergence testing only tests for convergence. A divergence test will never reveal if a serie...
- First, see if the given series fits any common forms of series. The series may be a geometric series, a p-series, or a telescoping sum. It the ser...
- Definition of p Series Test
- Historical Significance of p Series Test
- Properties
- Applications
- Exercise
The p-series testis a method used to determine the convergenceor divergenceof a specific type of series called the p-series. A p-seriesis defined as the sum of the terms (1/nᵖ) for n ranging from 1 to infinity. Mathematically, it can be represented as: ∑(1/nᵖ) In this representation, the symbol“∑” denotes the summation notation, “n” is the index va...
The historical significanceof the p-series testlies in its contribution to the development of mathematical analysis, particularly in the study of series convergence. While the test itself may not have a specific historical origin, its principles and applications have been explored by mathematicians over the centuries. Here’s a discussion on the his...
Specific to p-Series
The p-series test is specifically designed to analyze the convergence or divergence of the p-series of the form ∑(1/nᵖ). It is not applicable to other series or more general cases. This specialized nature ensures that the test is most effective when examining p-series.
Borderline Case
When the exponent “p” in the p-series is equal to 1, the series becomes the harmonic series ∑(1/n). In this case, the p-series test is inconclusive. The harmonic series neither converges nor diverges. It serves as a noteworthy example in the study of series convergence and is often discussed in relation to the p-series test.
Relationship to Other Tests
The p-series testhas a connection to other convergence tests, which allows for a more comprehensive understanding of series behavior. Two notable tests often used in conjunction with the p-series testare:
The p-series test, with its ability to determine the convergence or divergence of specific types of series, has found applications in various areas of mathematics and beyond. Here are some notable applications of the p-series test.
Example 1
Determine the convergence or divergence of the series ∑(1/n^3).
Solution
To analyze the convergence or divergence of the series, we can apply the p-series test with “p = 3”. Thep-series test states that if the exponent “p” is greater than1, the series converges; otherwise, it diverges. In this case, “p = 3” is greater than 1. Therefore, the series∑(1/n^3) converges.This implies that as more terms are added, the sum of the series approaches a finite value.
Example 2
Investigate the convergence or divergence of the series ∑(1/n⁰˙⁵).
Integral Test. If f ( n ) = a for all. and f (x ) is continuous, positive, and decreasing on [ 1 , ∞ ) , then: ∞ ∞. If ∫ f ( x ) dx converges, then ∑ a n converges. 1 n =1 ∞ ∞ ∑ If ∫ f ( x ) dx is divergent, then a is divergent. n. 1 n =1.
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This test gives us a quick way to determine if some series diverge. Determine if the series converges or diverges. Here, the sequence whose terms are being summed is given by the formula .
People also ask
What is the difference between a divergence test and a comparison test?
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May 10, 2023 · Use the \(n^{\text{th}}\) Term Test for Divergence to determine if a series diverges. Use the Integral Test to determine the convergence or divergence of a series. Estimate the value of a series by finding bounds on its remainder term.
