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duki
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Homework Statement
[tex]\sum \frac{(-1)^{n+1}(n^2+4)^{1/3}}{(n^5+1)^{1/2}}[/tex]
Homework Equations
The Attempt at a Solution
I got that it converges, because the limit is 0
Is that right?
The purpose of confirming convergence is to determine whether a given mathematical series will approach a definite limit or continue to increase indefinitely. This information is important in understanding the behavior and properties of the series.
To determine convergence, we can use various tests such as the ratio test, comparison test, or the root test. In this particular series, the ratio test can be used to confirm convergence by checking the limit of the absolute value of the ratio of consecutive terms.
The alternating signs in the series indicate that the terms are alternately positive and negative. This is important because it affects the overall behavior of the series and can have an impact on its convergence or divergence.
Yes, it is possible for a series to have terms that approach zero, but still diverge. This is because the rate at which the terms approach zero may not be fast enough to overcome the increasing terms in the series.
The power in the numerator and denominator can affect the convergence of the series by altering the rate at which the terms approach zero. In this series, the power of 1/3 in the numerator and 1/2 in the denominator indicates that the terms will approach zero at a slower rate, making it more likely for the series to converge.