--- title: "Geometric Series and the Sum to Infinity" description: "Sum a geometric sequence with the standard formula, and see that when the common ratio is between -1 and 1 the running total settles on a finite limit however many terms are added." canonical: https://lightmysky.com/learn/mathematics/geometric-series-and-the-sum-to-infinity-mt_HHxFOv6vXS source: https://lightmysky.com/learn/mathematics/geometric-series-and-the-sum-to-infinity-mt_HHxFOv6vXS.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Geometric Series and the Sum to Infinity Sum a geometric sequence with the standard formula, and see that when the common ratio is between -1 and 1 the running total settles on a finite limit however many terms are added. Subject: Mathematics · Area: Algebra · Ages 17 to 18 Page: https://lightmysky.com/learn/mathematics/geometric-series-and-the-sum-to-infinity-mt_HHxFOv6vXS ## Ready when they can - Find the sum of the first 12 terms of a geometric sequence - Decide from the common ratio whether a sum to infinity exists - Use a / (1 - r) on a recurring decimal or a repeated-dose model ## Lesson: Geometric sums that settle down A geometric sequence multiplies by the same ratio every step, which outruns anything adding a fixed amount. Find the ratio by dividing any term by the one before it, and check it works everywhere: 3, 6, 12 has ratio 2. Alternating signs just mean a negative ratio. The nth term is the first term times the ratio to the power n - 1, so check n = 1 gives the first term back. **Example.** Add finite runs with first term times (r^n - 1)/(r - 1). For 3, 6, 12 over 12 terms: 2^12 = 4096, so 3 times 4095 = 12285. For 2, 4, 8 over 6 terms: 2 times (64 - 1) = 126. An endless sum lands on a finite total only when its terms shrink fast, which happens exactly when the ratio sits strictly between -1 and 1. Then the r^n part melts to zero and first term/(1 - r) is left. So 1 + 1/2 + 1/4 + ... totals 1/(1 - 1/2) = 2. With a ratio of 2 or 1 the terms never shrink, so no finite total exists. **Example.** Recurring decimals are this formula in disguise. Write 0.777... as 0.7 + 0.07 + 0.007 + ..., a series with first term 0.7 and ratio 0.1, totalling 0.7/0.9 = 7/9. The same moves turn 0.454545... into 0.45 + 0.0045 + ..., with ratio 0.01, totalling 0.45/0.99 = 45/99 = 5/11. A tank getting 12 new units daily while keeping a quarter of its total settles at 12/(1 - 1/4) = 16. **Recap.** Find the ratio first, use it for finite sums, and trust first term/(1 - r) only when the ratio sits between -1 and 1. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Arithmetic Series and Sigma Notation](https://lightmysky.com/learn/mathematics/arithmetic-series-and-sigma-notation-mt_bAIPng87JO) - [Geometric Sequences and Common Ratios](https://lightmysky.com/learn/mathematics/geometric-sequences-and-common-ratios-mt_NDnbJm2hTY) ## Opens up - [The Molecular Partition Function](https://lightmysky.com/learn/science/the-molecular-partition-function-mt_-TPTlb7zQ2) - [Amortised Analysis](https://lightmysky.com/learn/computing/amortised-analysis-mt_1zLrl5RKGr) - [Recurrence Relations and the Master Theorem](https://lightmysky.com/learn/computing/recurrence-relations-and-the-master-theorem-mt_aOvLPV7rR8) - [Infinite Series and the Geometric Series](https://lightmysky.com/learn/mathematics/infinite-series-and-the-geometric-series-mt_EPE-ERFFQt) - [The Binomial Expansion](https://lightmysky.com/learn/mathematics/the-binomial-expansion-mt_z16HZkNZ8c)