10Unit 10: Infinite Sequences and Series (BC)

An infinite series is the limit of its partial sums.

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A geometric series multiplies by the same ratio each term.

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If the terms themselves do not shrink to zero, there is no way the running total can settle — the series must diverge.

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If the terms come from a positive, decreasing function, the series and the matching improper integral are like a staircase and a ramp — they rise together, so they converge or diverge together.

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A p-series $\sum 1/n^p$ converges exactly when the terms shrink fast enough — that threshold is $p>1$.

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Compare a messy positive series to a known one (usually a p-series or geometric).

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When terms alternate sign and steadily shrink to zero, the partial sums zig-zag inward toward a limit — like closing in from both sides.

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Look at how each term compares to the one before.

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A series can converge only because of sign cancellation (conditional) or robustly even if you make every term positive (absolute).

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Because an alternating series’ partial sums straddle the true sum, the error after stopping is no bigger than the first term you left out.

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A Taylor polynomial matches a function’s value and first several derivatives at a center, so it hugs the curve near that point — the more terms, the closer the fit.

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The error of a Taylor approximation is controlled by the size of the next derivative over the interval — it behaves like the first term you dropped.

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A power series converges only within a certain distance of its center — the radius.

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Letting a Taylor polynomial run to infinitely many terms gives a power series that equals the function on its interval of convergence.

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New series come cheaply from known ones: substitute, multiply, differentiate, or integrate a known series term by term to represent a related function.

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