Learn Construct upper and lower limits from entire tails, characterize subsequential behavior, and prove convergence tests without assuming ordinary root limits exist.
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Constructing endpoint subsequences requires approximate bounds, not an assumed largest term.
No. The values 1−1/n have supremum 1 without ever attaining it. A supremum supplies arbitrarily close values from below when finite.
Sparse or alternating roots may have no ordinary limit.
The ordinary root limit must exist. This unit replaces that premise with an upper limit and proves the resulting convergence and divergence cases.
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Construct the cluster set and audit a sparse power series
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Define limsup and liminf using infinite tails, construct their finite or infinite subsequential limits, and separate sampled bounds from eventual claims.
6 practice tasks.
Turn an eventual root bound into an absolute tail estimate, detect infinitely recurring large terms, and keep the boundary and sparse-index cases honest.
6 practice tasks.