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CalculusBy Educator Today

C42 · Unit C10 Sequences and series

Sequences, series and the nth-term test

BC extensionReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 10.1, 10.3 · BC
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

A sequence is a list a sub n. A series sum a sub n is studied through its partial sums S_N equals sum sub n ₌ sub 1 ᴺa sub n. The series converges when these partial sums approach a finite limit. If a sub n does not tend to zero, the series diverges. The converse is false: terms tending to zero do not ensure the partial sums settle.

Worked example

a sub n equals 1 over n tends to zero, but the harmonic series diverges. Group terms from n equals 2 to the power k +1 to 2^(k+1); each block has 2 to the power k terms at least 1 over 2^(k+1), so each block adds at least 1 over 2. Partial sums therefore grow without bound.

Explore the mathematics

Sequences, series and the nth-term 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.
Sequences, series and the nth-term test · visual worked method
Sequences, series and the nth-term test · visual worked methodaₙ=1/n tends to zero, but the harmonic series diverges. Group terms from n=2^k+1 to 2^(k+1); each block has 2^k terms at least 1/2^(k+1), so each block adds at least 1/2. Partial sums therefore grow without bound.Worked example · 1/2aₙ=1/n tends to zero, but theharmonic series diverges. Groupterms from n=2^k+1 to 2^(k+1);each block has 2^k terms atWorked example · 2/2least 1/2^(k+1), so each blockadds at least 1/2. Partial sumstherefore grow without bound.

aₙ=1/n tends to zero, but the harmonic series diverges. Group terms from n=2^k+1 to 2^(k+1); each block has 2^k terms at least 1/2^(k+1), so each block adds at least 1/2. Partial sums therefore grow without bound.

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

A common misconception

Always identify whether a question asks about the term sequence or the series. Their limits are different objects.

Investigate and explain

Plot both a sub n and S_N for 1 over n. Explain why a downward term plot can coexist with an upward unbounded sum plot.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.