- #1

abhi@maths

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**1. Find whether the following series converges or diverges or is oscillatory**

[tex]\sum1/(n^3*(sin^2 n))[/tex]

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- Thread starter abhi@maths
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In summary, the convergence of a series refers to the behavior of the sum of its terms as the number of terms increases. To determine if a series converges or diverges, various tests such as the comparison test, ratio test, root test, or integral test can be used. The series 1/(n^3*(sin^2 n)) converges, as proven by the comparison test. The convergence of this series has practical applications in fields such as science, engineering, and mathematics. In real-world applications, this series is used to model and analyze systems with oscillatory behavior.

- #1

abhi@maths

- 2

- 0

[tex]\sum1/(n^3*(sin^2 n))[/tex]

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- #2

nitro

- 9

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\sum (1/n^3) converges

sin^2 n is always positive!

try to combine these two facts together

sin^2 n is always positive!

try to combine these two facts together

- #3

Mene

- 1,656

- 0

Based on the given series, it is difficult to determine whether it converges or diverges or is oscillatory. Further information is needed to make a conclusion. Some possible approaches to determine the convergence of this series could include applying the comparison test, the ratio test, or the integral test. Additionally, the behavior of the sine function and its relationship to the terms in the series may also need to be considered. Without more information, it is not possible to provide a definitive answer.

The convergence of a series refers to the behavior of the sum of its terms as the number of terms increases. A series is said to converge if the sum of its terms approaches a finite value as the number of terms approaches infinity.

To determine if a series converges or diverges, one can use various tests such as the comparison test, ratio test, root test, or the integral test. These tests involve analyzing the behavior of the terms in the series as the number of terms increases.

The series 1/(n^3*(sin^2 n)) converges, as it can be proven using the comparison test. By comparing it to the series 1/n^3, which is a known convergent series, we can conclude that 1/(n^3*(sin^2 n)) also converges.

The convergence of this series has practical implications in various fields of science and engineering, such as in the analysis of electrical circuits and the study of oscillating systems. Understanding the convergence of this series can also aid in solving mathematical problems and equations that involve similar series.

Yes, the convergence of this series has real-world applications in fields such as signal processing, control theory, and physics. In these fields, the series 1/(n^3*(sin^2 n)) is used to model and analyze the behavior of systems that exhibit oscillatory behavior.

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