Series

A series is what you get by adding the terms of a sequence one after another, starting from the first. Whether that running total settles onto some value or keeps growing forever comes down to how quickly the terms shrink. For a geometric series the sum gathers to a single value only when the ratio has size below 1, and otherwise it diverges. Move the common ratio r and the term count n to see whether the partial-sum bars home in on a limit or shoot up without bound.

Adding Infinitely Many Terms

Number of partial-sum terms8

Common ratio r0.5

👀 See It

Gold bars are the partial sums Sₙ. When |r|<1 the bars approach the red dashed limit. When |r|≥1 they grow without bound (divergence).

Sense the Term Sizes

📦 Area Analogy

①Each square is one term rᵏ

②|r|<1: squares shrink fast, total area finite

③|r|≥1: squares do not shrink, total area infinite

Geometric Series Formula

Partial Sum

S_n = a(1 - rⁿ)1 - r (r ≠ 1)

nth partial sum of a geometric series with first term a and ratio r

Geometric Series (infinite)

a1 - r (|r| < 1)

As n → ∞, rⁿ → 0, so the partial sum converges to a/(1−r)

💡 Key Idea

①Sₙ = a(1−rⁿ)/(1−r); as n→∞, rⁿ→0

②So S = a/(1−r)

③Holds only when |r|<1!

Convergence Tests

Divergence Test

lim_n→∞ a_n ≠ 0 ⇒ Σa_n diverges

If the general term does not go to 0, the series must diverge

Comparison Test

0 ≤ a_n ≤ b_n : Σb_n converges ⇒ Σa_n converges

If a larger series converges, the smaller one does too (and the converse for divergence)

⚖️ Test Analogies

①Divergence test: if terms do not approach 0, the sum cannot stop growing

②Comparison: if the bigger series passes, the smaller does too

Worked Examples

Example 1

Find the sum of the infinite geometric series Σ_n=1^∞ (2/3)ⁿ.

Identify the first term a and the common ratio r.

a = 23, r = 23 (|r| < 1)

Since |r| < 1, apply S = a/(1−r).

S = 2/31 - 2/3 = 2/31/3 = 2

▸ 2

For a geometric series, pin down the first term and ratio, then plug straight into the sum formula.

Example 2

Determine whether Σ_n=1^∞ n2n+1 converges or diverges.

Before summing, check the limit of the general term.

lim_n→∞ n2n+1 = 12

The term does not go to 0, so by the divergence test the series diverges.

lim a_n = 12 ≠ 0 ⇒ diverges

▸ Diverges

Always check whether the general term tends to 0 first. If not, the series diverges immediately.

Wrap-up

Geometric Convergence

Σ_k=0^∞ ar^k = a1-r (|r| < 1)

Converges when |r|<1

CSAT-style

A geometric series with first term a and ratio r satisfies Σ_n=1^∞ ar^n-1 = 9 and Σ_n=1^∞ a² r^2(n-1) = 812. Find a.

①4

②5

③6

④7

⑤8

▸ ③ 6

Write each sum (the second series has ratio r²).

a1-r = 9, a²1-r² = 812

Substitute a = 9(1−r) and simplify to fix r and a.

1-r1+r = 12 ⇒ r = 13, a = 6

🎯 Exam Points

①Geometric: |r|<1 converges, |r|≥1 diverges

②Infinite sum: a/(1−r)

③Divergence test: lim aₙ ≠ 0 ⇒ divergent

④Series convergence is defined as the limit of Sₙ

⑤Comparison: bigger converges → smaller converges

← Previous

Limit of a Geometric Sequence

Infinite Geometric Series

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