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Unit 10
15–20% of examThis BC-only unit asks a surprising question: can you add up infinitely many numbers and get a finite answer? You learn tests that decide whether a series converges, then use Taylor and power series to write functions like eˣ and sin x as never-ending polynomials. You also learn to measure how far off an approximation can be.
Longer videos that cover the whole unit. Good for a first pass or a final review.
An infinite series adds the terms of a sequence forever. You look at the partial sums S₁, S₂, S₃, …; if they approach a finite number, the series converges to that number, and if not, it diverges.
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In a geometric series, each term is the previous one times the same ratio r. It converges exactly when |r| < 1, and then its sum is a/(1 − r), where a is the first term.
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If the terms of a series don't approach 0, the series diverges. If the terms do approach 0, this test tells you nothing, and you need a different test.
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If the terms come from a function f that is positive, continuous and decreasing, the series and the improper integral of f either both converge or both diverge. The integral's value isn't the series' sum, though; it only tells you whether the series converges.
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A p-series adds up 1/nᵖ; it converges when p > 1 and diverges when p ≤ 1. The harmonic series, 1 + 1/2 + 1/3 + …, is the case p = 1, and it diverges even though its terms go to 0.
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Comparison tests judge a series with positive terms against one you already know, often a geometric series or p-series. With direct comparison, a series smaller than a convergent one converges and a series bigger than a divergent one diverges; with limit comparison, if the ratio of the terms approaches a positive, finite number, both series do the same thing.
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An alternating series switches between positive and negative terms. It converges if the sizes of the terms keep decreasing and approach 0, which is why 1 − 1/2 + 1/3 − 1/4 + … converges.
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The ratio test finds the limit L of |aₙ₊₁/aₙ|. If L < 1 the series converges (absolutely), if L > 1 it diverges, and if L = 1 the test gives no answer; it works especially well with factorials and powers like 2ⁿ.
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A series converges absolutely if the series of its absolute values converges, and that guarantees the original series converges too. If a series converges but the series of absolute values diverges, it converges conditionally, like the alternating harmonic series.
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If an alternating series passes the alternating series test and you stop after some number of terms, your error is no bigger than the size of the first term you left out.
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A Taylor polynomial matches a function's value and its first few derivatives at one point x = a, so it closely follows the function near a. The coefficient of (x − a)ⁿ is f⁽ⁿ⁾(a)/n!, and a Taylor polynomial centered at 0 is called a Maclaurin polynomial.
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The Lagrange error bound caps how far a degree-n Taylor polynomial can be from the true value. The error is at most M·|x − a|ⁿ⁺¹/(n + 1)!, where M is the largest value of |f⁽ⁿ⁺¹⁾| between a and x.
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A power series converges for x-values within some distance R of its center, called the radius of convergence, and you usually find R with the ratio test. Check each endpoint separately, since the series might converge at both, one or neither.
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A Taylor series is the never-ending version of a Taylor polynomial. Know the Maclaurin series for eˣ, sin x, cos x and 1/(1 − x) well, because many other series are built from them.
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You can make new power series from known ones by substituting, multiplying by a power of x, or differentiating or integrating term by term. Differentiating or integrating keeps the same radius of convergence, but the series may behave differently at the endpoints.
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