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Sum to Infinity

Q: How do you calculate Sum to Infinity?

If |r|<1, the terms shrink towards 0 and the geometric series converges to a finite sum.

Q: What are common mistakes with the Sum to Infinity formula?

Using for r > 1. Forgetting condition |r| < 1.

Q: What are some study tips for the Sum to Infinity formula?

Always verify that |r| < 1 before applying the formula. Identify the first term 'a' as the value of the sequence at index zero. Ensure the series is geometric by checking if the ratio between terms is constant.

Limit of a geometric series as n approaches infinity.

Understand the formulaSee the free derivationOpen the full walkthrough

S_{\infty} = \frac{a}{1 - r}

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Core idea

Overview

This formula represents the limit of the sum of a geometric series as the number of terms approaches infinity. It defines the point of convergence for a sequence where each successive term is generated by multiplying the previous one by a constant ratio.

When to use: This equation is applicable only to geometric progressions where the absolute value of the common ratio is strictly less than one. If the ratio is outside the range of -1 to 1, the series diverges and the sum is undefined.

Why it matters: Infinite sums are vital in financial modeling for determining the present value of perpetuities and in computer science for analyzing recursive algorithms. They also appear in physics when calculating the total distance traveled by bouncing objects or analyzing wave patterns.

Symbols

Variables

S_{\infty} = Sum to Infinity, a = First Term, r = Common Ratio

S_{\infty}

Sum to Infinity

V a r iab l e

a

First Term

V a r iab l e

r

Common Ratio

(∣ r ∣ < 1)

Walkthrough

Derivation

Derivation of Sum to Infinity for a Geometric Series

If |r|<1, the terms shrink towards 0 and the geometric series converges to a finite sum.

The series is geometric.
|r|<1.

Start from the Finite Sum Formula:

Use the formula for the sum of the first n terms.

S_{n} = \frac{a ( 1 - r ^{n} )}{1 - r}

Use the Convergence Condition:

A number with magnitude less than 1 tends to 0 when raised to large powers.

n \to \infty lim r^{n} = 0 (∣ r ∣ < 1)

Take the Limit:

Substitute $r^{n} \to 0$ to obtain the sum to infinity.

S_{\infty} = \frac{a}{1 - r}

Result

S_{\infty} = \frac{a}{1 - r}

Source: AQA A-Level Mathematics — Pure (Sequences and Series)

Free formulas

Rearrangements

Solve for $S_{\infty}$

Make $S_{\infty}$ the subject

S_{\infty} = \frac{a}{1 - r}

The formula is already rearranged to make $S_{\infty}$ the subject.

Difficulty: 1/5

Solve for $a$

Make a the subject

a = S_{\infty} (1 - r)

Rearranges the sum to infinity formula to solve for the first term.

Difficulty: 2/5

Solve for $r$

Make r the subject

r = 1 - \frac{a}{S _{\infty}}

Rearranges the sum to infinity formula to solve for the common ratio.

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

The graph of the sum to infinity is a hyperbolic curve plotted with the common ratio on the x-axis and the sum on the y-axis. As the ratio approaches one, the graph exhibits a vertical asymptote, while the y-intercept is defined by the first term a.

Graph type: hyperbolic

Why it behaves this way

Intuition

Visualize a bouncing ball that loses a fixed fraction of its height with each bounce; the sum to infinity represents the total vertical distance it travels before coming to rest.

S_{\infty}

The finite sum that an infinite geometric series approaches when it converges.

Represents the ultimate total value accumulated by adding an endless sequence of shrinking terms.

The value of the first term in the geometric series.

Establishes the starting magnitude or initial contribution to the total sum.

The constant ratio by which each term is multiplied to obtain the next term in the series.

Dictates the rate at which terms decrease in magnitude; if |r| < 1, terms shrink, allowing the sum to converge.

Signs and relationships

1-r: The term 1-r in the denominator is fundamental to the convergence of the series. If |r| < 1, then 1-r will be a positive value between 0 and 2.

Free study cues

Insight

Canonical usage

The sum to infinity, $S_{i}$ nfinity, carries the same units as the first term, a, while the common ratio, r, must be dimensionless.

Common confusion

A common mistake is to assign units to the common ratio 'r', or to assume $S_{i}$ nfinity is dimensionless when 'a' has units.

Dimension note

The common ratio 'r' is inherently dimensionless, as it is derived from the ratio of successive terms (e.g., $a_{2}$ / $a_{1}$ ). If $a_{1}$ and $a_{2}$ have units, those units must cancel.

Unit systems

$S_{\infty}$ Same as 'a' · The sum to infinity will have the same units and dimension as the first term, 'a'.

$a$ Any consistent unit · The first term of the geometric series. Its dimension determines the dimension of S_infinity.

$r$ dimensionless · The common ratio must be a pure number without units for the series to be well-defined and for the formula to hold.

One free problem

Practice Problem

A geometric series starts with 100 and each subsequent term is half of the one before it. Calculate the total sum of this series as it continues to infinity.

First Term100

Common Ratio0.5 (|r|<1)

Solve for: $S$

Hint: Identify the first term as 100 and the ratio as 0.5, then substitute into the formula.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Zeno's paradox.

Study smarter

Tips

Always verify that |r| < 1 before applying the formula.
Identify the first term 'a' as the value of the sequence at index zero.
Ensure the series is geometric by checking if the ratio between terms is constant.

Avoid these traps

Common Mistakes

Using for r > 1.
Forgetting condition |r| < 1.

Common questions

Frequently Asked Questions

If |r|<1, the terms shrink towards 0 and the geometric series converges to a finite sum.

This equation is applicable only to geometric progressions where the absolute value of the common ratio is strictly less than one. If the ratio is outside the range of -1 to 1, the series diverges and the sum is undefined.

Infinite sums are vital in financial modeling for determining the present value of perpetuities and in computer science for analyzing recursive algorithms. They also appear in physics when calculating the total distance traveled by bouncing objects or analyzing wave patterns.

Using for r > 1. Forgetting condition |r| < 1.

Zeno's paradox.

Always verify that |r| < 1 before applying the formula. Identify the first term 'a' as the value of the sequence at index zero. Ensure the series is geometric by checking if the ratio between terms is constant.

References

Sources

Wikipedia: Geometric series
Britannica: Geometric series
Stewart, James. Calculus: Early Transcendentals. Cengage Learning.
AQA A-Level Mathematics — Pure (Sequences and Series)

Sum to Infinity

Overview

Variables

Derivation

Start from the Finite Sum Formula:

Use the Convergence Condition:

Take the Limit:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Geometric Series (Sum)

Frequently Asked Questions

Sources