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ineedhelpnow
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comparison or root test? for testing convergence/divergence $\sum_{k=1}^{\infty}\frac{(-3)^{k+1}}{4^{2k}}$
ineedhelpnow said:comparison or root test? for testing convergence/divergence $\sum_{k=1}^{\infty}\frac{(-3)^{k+1}}{4^{2k}}$
The Root Test/Comparison is a method used to determine the convergence or divergence of an infinite series. It involves taking the nth root of the absolute value of the terms in the series and then evaluating the limit as n approaches infinity. If the limit is less than 1, the series converges. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive and another method must be used.
To apply the Root Test/Comparison to this series, we first take the nth root of the absolute value of the terms, which gives us $\sqrt[n]{\frac{|(-3)^{k+1}|}{|4^{2k}|}}$. Simplifying this, we get $\frac{3}{4^{2k/n}}$. Then, we take the limit as n approaches infinity, which gives us $\frac{3}{4^2}= \frac{3}{16}$. Since this limit is less than 1, the series converges.
No, the Root Test/Comparison can only be used to determine the convergence or divergence of series that have positive terms. If the terms of the series are not all positive, another method, such as the Alternating Series Test or the Ratio Test, must be used.
Yes, the Root Test/Comparison is a reliable method for determining convergence or divergence. However, it should be noted that the test may sometimes be inconclusive, in which case another method must be used.
Yes, the Root Test/Comparison can be used in conjunction with the Ratio Test. In some cases, using both tests may provide a more conclusive result than using either test alone.