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CalculusBy Educator Today

C43 · Unit C10 Sequences and series

Geometric and p-series; the integral test

BC extensionReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 10.2, 10.4–10.5 · BC
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

A geometric series a+ar+ar squared + and so on converges to a/(1 minus r) when |r|<1. For |r| is greater than or equal to 1 and a is not equal to 0 it diverges. The p-series sum 1 over nᵖ converges exactly for p>1. The integral test compares a series with ∫f when f(n) equals a sub n and f is eventually positive, continuous and decreasing. It decides convergence, not equality of the series and integral values.

Worked example

sum sub n ₌ sub 0 ∞3(1 over 4) to the power n equals 3/(1 minus 1 over 4) equals 4. For sum sub n ₌ sub 1 ∞1 over n squared, the integral ∫ sub 1 ∞x to the power minus 2 dx is finite, so the series converges; its sum is not 1 merely because the integral is 1.

Explore the mathematics

Geometric and p-series; the integral test

Compare finite partial sums with a convergent geometric limit.

Setting 3 of 5

Scroll sideways to see the full diagram.

Predict the next term and how the partial sum changes.1uₙ=(0.5)ⁿ⁻¹; first 1 terms

Predict before continuing

Predict the next term and how the partial sum changes.

Read the complete explanation
  1. Predict the next term and how the partial sum changes.
  2. Add a new term; a sequence lists terms, whereas a series adds them.
  3. The displayed partial sum is 1.
  4. Because |r|<1, the infinite sum is 1/(1−r)=2. A finite partial sum is not yet the limit. Terms tending to zero are necessary but not sufficient for a general series to converge.
Geometric and p-series; the integral test · visual worked method
Geometric and p-series; the integral test · visual worked methodΣₙ₌₀∞3(1/4)ⁿ=3/(1−1/4)=4. For Σₙ₌₁∞1/n², the integral ∫₁∞x⁻²dx is finite, so the series converges; its sum is not 1 merely because the integral is 1.Worked example · 1/2Σₙ₌₀∞3(1/4)ⁿ=3/(1−1/4)=4. ForΣₙ₌₁∞1/n², the integral ∫₁∞x⁻²dxis finite, so the seriesconverges; its sum is not 1 merelyWorked example · 2/2because the integral is 1.

Σₙ₌₀∞3(1/4)ⁿ=3/(1−1/4)=4. For Σₙ₌₁∞1/n², the integral ∫₁∞x⁻²dx is finite, so the series converges; its sum is not 1 merely because the integral is 1.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

The first term depends on the starting index. Check it before applying the geometric sum formula.

Investigate and explain

Compare a decreasing curve’s rectangles and improper integral. Derive upper and lower estimates for the series tail.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.