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The nth Term Test for Divergence is a crucial tool in calculus for determining the divergence of infinite series. It states that if the limit of the sequence of terms does not approach zero as n approaches infinity, the series diverges. This test is vital for initial series analysis but cannot confirm convergence. Examples like the Harmonic series and P-series illustrate its application, while the Integral Test offers a more comprehensive analysis for both convergence and divergence.
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The \(n^{th}\) Term Test for Divergence is a method used to determine if a series diverges by examining the limit of the sequence of terms
Belief that approaching zero guarantees convergence
A common misconception is that if the terms of a series approach zero, the series must converge, but this is not always the case
Limit of sequence must be zero for convergence
Another misconception is that the limit of the sequence of terms must be zero for a series to converge, but this is only a necessary condition, not a sufficient one
The \(n^{th}\) Term Test for Divergence is based on the contrapositive that if a series converges, the limit of the sequence of terms must be zero
The Integral Test is a method used to determine the convergence or divergence of a series by comparing it to an improper integral
The Integral Test can be applied to series with terms from a positive, continuous, and monotonically decreasing function
Unlike the \(n^{th}\) Term Test for Divergence, the Integral Test can confirm both convergence and divergence of a series
00
If a series converges, the limit of its terms as they approach ______ must be ______.
infinity
zero
01
If a function generating a series' terms is positive, continuous, and ______ decreasing, the ______ Test can be applied.
monotonically
Integral
