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squenshl
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Do I use the comparison test or limit comparison test to see if the series sin^2(1/n) converges or diverges and if the series n^2/(n^2+1) converges or diverges.
A series converges if the sum of its terms approaches a finite value as the number of terms increases. In other words, the terms of the series get smaller and smaller, eventually approaching zero.
There are several tests that can be used to determine if a series converges, such as the ratio test, the root test, and the integral test. These tests involve analyzing the behavior of the terms in the series and comparing them to known patterns of convergence or divergence.
If a series does not converge, it is said to diverge. This means that the sum of its terms does not approach a finite value, and the series may either oscillate between positive and negative values or grow infinitely large.
Yes, a series can converge to a negative value. This means that the sum of its terms approaches a finite negative value as the number of terms increases. However, it is more common for series to converge to a positive value or to have no convergence at all.
Determining if a series converges is important in many areas of science and mathematics, as it allows us to make accurate predictions and calculations. For example, in physics, series convergence is used in calculating electric and magnetic fields. In economics, it is used in calculating interest rates and growth rates. In general, understanding series convergence helps us make sense of the world around us and make informed decisions based on mathematical principles.