Root Test for for Convergence of an Infinite Series
Key Questions
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I would use Root Test when the terms of the series are in the form of some expression to the nth power; otherwise, I would try other tests first.
Example
Let us look at examine the convergence of the series:
#sum_{n=1}^infty({2n}/{5-3n})^n# By Root Test,
#lim_{n to infty}root{n}{|({2n}/{5-3n})^n|}=lim_{n to infty}|{2n}/{5-3n}|# by dividing the numerator and the denominator by
#n# ,#=lim_{n to infty}|{2}/{5/n-3}|=|{2}/{0-3}|=2/3<1# Hence, the series is absolutely convergent.
I hope that this was helpful.
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Let
#a_n=({n^2+1]/{2n^2+1})^n# .By Root Test,
#lim_{n to infty}root[n]{|a_n|}=lim_{n to infty}root[n]{|({n^2+1}/{2n^2+1})^n|}# by cancelling out the nth-root and the nth-power,
#=lim_{n to infty}{n^2+1}/{2n^2+1}# (Note: the absolute value is not necessary since it is already positive.)
by dividing by
#n^2# ,#=lim_{n to infty}{1+1/n^2}/{2+1/n^2}={1+0}/{2+0}=1/2<1# Hence, the series converges.
I hope that this was helpful.
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Root Test
If
#lim_{n to infty}root[n]{|a_n|}<1# , then#sum_{n=1}^inftya_n# converges.
If#lim_{n to infty}root[n]{|a_n|}>1# , then#sum_{n=1}^inftya_n# diverges.
If#lim_{n to infty}root[n]{|a_n|}=1# , then it is inconclusive.
I hope that this was helpful.
Questions
Tests of Convergence / Divergence
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Geometric Series
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Nth Term Test for Divergence of an Infinite Series
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Direct Comparison Test for Convergence of an Infinite Series
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Ratio Test for Convergence of an Infinite Series
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Integral Test for Convergence of an Infinite Series
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Limit Comparison Test for Convergence of an Infinite Series
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Alternating Series Test (Leibniz's Theorem) for Convergence of an Infinite Series
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Infinite Sequences
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Root Test for for Convergence of an Infinite Series
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Infinite Series
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Strategies to Test an Infinite Series for Convergence
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Harmonic Series
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Indeterminate Forms and de L'hospital's Rule
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Partial Sums of Infinite Series