IB Mathematics Analysis and Approaches Topic SL 1.8…

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IB Mathematics Analysis and Approaches Topic SL 1.8… | RevisionDojo

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Can An Infinite Sum Be Finite?

It sounds impossible to add infinitely many numbers and get a finite answer, but it can happen if the terms shrink fast enough.
Think of $\tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} + \dots$: each step adds half of what is left, so the total creeps toward 1 without ever passing it.
For a geometric series this works exactly when the common ratio satisfies $|r| < 1$, so every term is a fraction of the one before.

Deriving The Sum To Infinity

Start from the finite sum formula from SL 1.3 and see what happens as the number of terms grows. $$ S_n = \frac{u_1(1 - r^{n})}{1 - r} $$
When $|r| < 1$, repeatedly multiplying by $r$ shrinks the result toward zero, so $r^{n} \to 0$ as $n \to \infty$.
1. For instance with $r = \tfrac{1}{2}$ the powers are $\tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{8}, \dots$, heading to 0.
Replacing $r^{n}$ with 0 in the finite formula gives the sum to infinity. $$ S_\infty = \frac{u_1(1 - 0)}{1 - r} = \frac{u_1}{1 - r}, \quad |r| < 1 $$

Exam technique

The formula booklet (SL 1.8) gives $S_\infty = \dfrac{u_1}{1-r}$ together with the condition $|r| < 1$.
Always check $|r| < 1$ before you use it; if that fails, there is no finite sum to find.

Why The Condition On r Is Essential

If $|r| \geq 1$ the terms do not shrink, so the running total keeps growing and the series diverges with no finite sum.
The condition uses the modulus $|r|$ because a negative ratio with $|r| < 1$ still converges; the terms just alternate in sign as they shrink.

Example

$2 + 4 + 8 + \dots$ has $r = 2$, so it diverges and has no infinite sum.
$8 - 4 + 2 - \dots$ has $r = -\tfrac{1}{2}$, so $|r| < 1$ and it converges to $\dfrac{8}{1-(-\frac12)} = \dfrac{16}{3}$.

Application: Turning Recurring Decimals Into Fractions

A recurring decimal is secretly an infinite geometric series, which is how you convert one into an exact fraction.

Example

$$0.4444\dots = \tfrac{4}{10} + \tfrac{4}{100} + \tfrac{4}{1000} + \dots$$ with $u_1 = \tfrac{4}{10}$ and $r = \tfrac{1}{10}$, so its sum is $\dfrac{4/10}{1 - 1/10} = \dfrac{4}{9}$.

Example question

An infinite geometric series is $12 + 4 + \tfrac{4}{3} + \dots$. Find its sum to infinity.

Solution

Find $r$ by dividing consecutive terms: $$r = \tfrac{4}{12} = \tfrac{1}{3}$$ and $|r| < 1$, so a finite sum exists. $$S_\infty = \dfrac{u_1}{1-r} = \dfrac{12}{1 - \frac13} = \dfrac{12}{\frac23} = 18$$

Active recall

State the formula for the sum to infinity and its condition.
Why does $r^n \to 0$ when $|r| < 1$, and what does that do to the finite sum formula?
Does $5 + 5 + 5 + \dots$ have a sum to infinity? Explain.
Why does the condition use $|r|$ rather than just $r$?
Write $0.5555\dots$ as an exact fraction using a geometric series.

SL 1.7 Laws of exponents and logs

SL 1.9 Binomial theorem where n is an integer

Can An Infinite Sum Be Finite?

It sounds impossible to add infinitely many numbers and get a finite answer, but it can happen if the terms shrink fast enough.
Think of $\tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} + \dots$: each step adds half of what is left, so the total creeps toward 1 without ever passing it.
For a geometric series this works exactly when the common ratio satisfies $|r| < 1$, so every term is a fraction of the one before.

Deriving The Sum To Infinity

Start from the finite sum formula from SL 1.3 and see what happens as the number of terms grows. $$ S_n = \frac{u_1(1 - r^{n})}{1 - r} $$
When $|r| < 1$, repeatedly multiplying by $r$ shrinks the result toward zero, so $r^{n} \to 0$ as $n \to \infty$.
1. For instance with $r = \tfrac{1}{2}$ the powers are $\tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{8}, \dots$, heading to 0.
Replacing $r^{n}$ with 0 in the finite formula gives the sum to infinity. $$ S_\infty = \frac{u_1(1 - 0)}{1 - r} = \frac{u_1}{1 - r}, \quad |r| < 1 $$

Exam technique

The formula booklet (SL 1.8) gives $S_\infty = \dfrac{u_1}{1-r}$ together with the condition $|r| < 1$.
Always check $|r| < 1$ before you use it; if that fails, there is no finite sum to find.

Why The Condition On r Is Essential

If $|r| \geq 1$ the terms do not shrink, so the running total keeps growing and the series diverges with no finite sum.
The condition uses the modulus $|r|$ because a negative ratio with $|r| < 1$ still converges; the terms just alternate in sign as they shrink.

Example

$2 + 4 + 8 + \dots$ has $r = 2$, so it diverges and has no infinite sum.
$8 - 4 + 2 - \dots$ has $r = -\tfrac{1}{2}$, so $|r| < 1$ and it converges to $\dfrac{8}{1-(-\frac12)} = \dfrac{16}{3}$.

Application: Turning Recurring Decimals Into Fractions

A recurring decimal is secretly an infinite geometric series, which is how you convert one into an exact fraction.

Example

$$0.4444\dots = \tfrac{4}{10} + \tfrac{4}{100} + \tfrac{4}{1000} + \dots$$ with $u_1 = \tfrac{4}{10}$ and $r = \tfrac{1}{10}$, so its sum is $\dfrac{4/10}{1 - 1/10} = \dfrac{4}{9}$.

Example question

An infinite geometric series is $12 + 4 + \tfrac{4}{3} + \dots$. Find its sum to infinity.

Solution

Active recall

State the formula for the sum to infinity and its condition.
Why does $r^n \to 0$ when $|r| < 1$, and what does that do to the finite sum formula?
Does $5 + 5 + 5 + \dots$ have a sum to infinity? Explain.
Why does the condition use $|r|$ rather than just $r$?
Write $0.5555\dots$ as an exact fraction using a geometric series.

IB Mathematics Analysis and Approaches SL 1.8 Sum of Infinite Geo Sequence Notes

Can An Infinite Sum Be Finite?

Deriving The Sum To Infinity

Why The Condition On r Is Essential

Application: Turning Recurring Decimals Into Fractions

IB Mathematics Analysis and Approaches SL 1.8 Sum of Infinite Geo Sequence Notes

Can An Infinite Sum Be Finite?

Deriving The Sum To Infinity

Why The Condition On r Is Essential

Application: Turning Recurring Decimals Into Fractions

Number and Algebra

Functions

Geometry & Trigonometry

Statistics & Probability

Calculus

Number and Algebra

Functions

Geometry & Trigonometry

Statistics & Probability

Calculus

IB Mathematics Analysis and Approaches SL 1.8 Sum of Infinite Geo Sequence Notes

Can An Infinite Sum Be Finite?

Deriving The Sum To Infinity

Why The Condition On r Is Essential

Application: Turning Recurring Decimals Into Fractions

IB Mathematics Analysis and Approaches SL 1.8 Sum of Infinite Geo Sequence Notes

Can An Infinite Sum Be Finite?

Deriving The Sum To Infinity

Why The Condition On r Is Essential

Application: Turning Recurring Decimals Into Fractions