Infinite Geometric Series

Adding Forever, Yet It Ends?

Terms added n4

👀 See It

①Adding more terms grows the bar

②But if the ratio is below 1, each added amount shrinks

③So even adding forever, it never passes a certain value (the red line) and converges to it

Start From the Partial Sum

Partial sum of a geometric sequence

S_n = a(1 - rⁿ)1 - r (r ≠ 1)

Sum of the first n terms with first term a and ratio r

🧱 First, the Sum of n Terms

①An infinite series is defined as the limit of partial sums Sₙ

②Write the geometric partial-sum formula first

③Then send n→∞ — this order is the key

Take the Limit and the Formula Appears

Sum of an infinite geometric series

S = a1 - r (|r| < 1)

With first term a and ratio r, it converges only if |r|<1, with sum a/(1−r)

Convergence condition

converges ⇔ a = 0 or |r| < 1

The first term is 0, or the ratio has absolute value below 1

💨 rⁿ Vanishes

①If |r|<1, then rⁿ→0 as n→∞

②The rⁿ term disappears from the partial sum

③What remains is the infinite-geometric-series formula

Compute It Directly

Example 1

Find the sum of the infinite geometric series with first term 3 and ratio 1/3.

Check the ratio — since |1/3|<1, it converges.

Substitute a=3, r=1/3 into S = a/(1−r).

S = 31 - 1/3 = 32/3

▸ S = 9/2

Checking |r|<1 convergence before applying the formula prevents mistakes.

Wrap-up

Key result

|r|<1 ⇒ ∑_n=1^∞ a r^n-1 = a1 - r

An infinite geometric series with first term a, ratio r converges to a/(1−r) when |r|<1

2023 KICE mock exam Math (Calculus) type, adapted

Find ∑_n=1^∞ 2ⁿ3ⁿ.

①1

②3/2

③2

④5/2

⑤Diverges

▸ ③ 2

Reading the general term as (2/3)ⁿ, it is a geometric series with first term 2/3 and ratio 2/3.

Since |2/3|<1, S = (2/3)/(1−2/3).

S = 2/31/3 = 2

🎯 Exam Points

①Always check the |r|<1 convergence condition first

②The sum is S=a/(1−r); pin down the first term and ratio exactly

③Mind the first-term difference between ∑arⁿ and ∑arⁿ⁻¹

④Repeating decimals and figure sums reduce to this formula too

⑤If it does not converge, the sum does not exist

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