When the ratio is small enough, an endless sum still settles on a number

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What to assume they know

Subtracting a scaled copy of the total to collapse it to two terms — a G.P.'s total over its opening n entries, and the derivation by subtracting a scaled copy
What changes once the terms are added instead of listed — the distinction between a series and its sum, and the fact that a series may be infinite
A constant ratio between neighbours is the entire definition — the constant-ratio definition and the non-zero condition
Absolute value of a signed number
Reading a small decimal with several leading zeros, and comparing two of them
Laws of exponents, for the item that turns a product of powers into a sum of exponents

What they should be able to do

Explain what it means for an endless addition to have a sum, in terms of the finite totals rather than in terms of adding forever
Split a geometric progression's n-term total into a part independent of n and a part depending on n
State the condition on the common ratio under which the second part vanishes, and say why it is needed
Compute the sum to infinity of a given G.P., with a positive or a negative ratio
Explain what the printed numerical table does and does not establish
Identify a G.P. for which no sum to infinity exists, and say what fails
Convert a product of powers into a sum-to-infinity problem in the exponents
Solve a sum-to-infinity problem where the answer is required in terms of two other infinite sums

Where it usually goes wrong

"Adding endlessly many positive numbers must give an endless answer." The chapter's own first instance adds forever and stays under 3. Show the finite totals climbing and flattening before any formula appears.
"The sum to infinity is an approximation, or the value it nearly reaches." No finite total equals it — for the worked instance every finite total falls short of 3 — and the sum to infinity is nonetheless exactly 3. The number is defined by what the totals approach, not by any one of them.
**"You get it by substituting infinity for n."** You cannot substitute infinity for anything. What happens is that one piece of the expression is driven towards zero and the rest of the expression is left alone.
"The condition means the ratio is a positive fraction." It is a condition on size. The section's own second example runs on a ratio of −1/2, and one of the exercise items runs on −1/4.
"This is a new formula to memorise alongside the finite one." It is the finite formula with the shrinking part removed. Derive it from the finite one every time.
"The table proves it." Four computed values make the claim believable and establish nothing. The proof is the split on the following page, and the distinction between evidence and argument is worth naming explicitly here.
"If the terms shrink towards zero the total must settle." Within geometric progressions the shrinking of the terms and the shrinking of the tail are the same fact, so the reasoning holds — but this chapter gives no basis for the claim about sequences in general.
"A ratio of zero would be the easiest case of all." A G.P. requires every term to be non-zero, so a ratio of zero was excluded back in §8.4 and never arises here.

Questions to check understanding

Compute the sum to infinity of a given G.P., with the ratio identified first
Decide whether a given G.P. has a sum to infinity, with a reason
Express a repeating decimal or a product of powers as an endless G.P. and total it
Given the sum to infinity and the first term, recover the common ratio
Solve a problem in which one endless total is required in terms of others
Explain, in words, why the size condition is needed

Examples worth working on the board

Values marked verified are worked out here on the printed inputs.

Where this material actually is. §8.6 is not printed in the chapter file, which ends at §8.5 on p. 145 and then runs to its Miscellaneous material and Summary. The section appears in the book's Supplementary Material at pp. 357–358, headed with the chapter number, and it carries its own Exercise 8.3. Anyone building the explanation from the chapter alone will not find the content. Say the location once, plainly, and move on.
The worked instance (§8.6, p. 357). Inputs: the G.P. beginning 1, then 2/3, then 4/9. With a = 1 and r = 2/3 the total of the first n terms is 3 times the quantity 1 − (2/3)ⁿ. Verified: substituting a and r into the finite formula gives exactly that, since 1 − 2/3 is 1/3 and dividing by it multiplies by 3.
The table (§8.6, p. 357). The page tabulates (2/3)ⁿ at n = 1, 5, 10 and 20, printing 0.6667, then 0.1316872428, then 0.01734152992, then 0.00030072866. Verified: (2/3)⁵ is 32/243, (2/3)¹⁰ is 1024/59049, and (2/3)²⁰ is the square of that; all three printed decimals agree with those fractions. Note for production: the four entries are given to four, ten, ten and eight significant figures respectively, so the row is inconsistently presented — do not imply the four columns were computed to the same precision.
The conclusion drawn from it (§8.6, p. 357). The section says that as n grows the quantity gets close to zero, and hence that the endless addition comes to 3. Verified: the finite total is 3 less three times a positive quantity, so every finite total is under 3 and none of them equals 3. That observation is added here and it is worth making — it is the sharpest available answer to a student who asks whether the sum is "really" 3.
The general argument (§8.6, p. 358). For a G.P. whose ratio has size under one, the section rewrites the n-term total as a ÷ (1 − r) minus arⁿ ÷ (1 − r), states that rⁿ goes to zero because of the size condition, and concludes that the totals approach a ÷ (1 − r). The symbol for the result is written with an infinity subscript, or as plain S. The split is the whole argument; an explanation that shows only the final formula has skipped the topic.
The two printed examples (§8.6, p. 358). Inputs: the endless addition of 1, 1/2, 1/4, 1/8, and onward; and the endless addition of 1, then −1/2, then 1/4, then −1/8, and onward. Verified: the first has a = 1 and r = 1/2, and a ÷ (1 − r) is 2; the second has a = 1 and r = −1/2, and the denominator becomes 1 + 1/2, so the answer is 2/3. The second is the one to dwell on — the condition is about the size of the ratio, so a negative ratio is admitted, and the addition that alternates in sign still settles.
Exercise 8.3 inputs (§8.6, p. 358). The G.P.s whose endless totals are wanted: 1, 1/3, 1/9, and onward; 6, 1.2, 0.24, and onward; 5, 20/7, 80/49, and onward; and −3/4, 3/16, −3/64, and onward. Verified by working added here: the ratios are 1/3, 0.2, 4/7 and −1/4, giving totals of 3/2, 7.5, 35/3 and −3/5 respectively. The remaining two items are arguments rather than computations: one asks for a proof that the endless product of 3 raised to 1/2, then 1/4, then 1/8, and onward equals 3; the other sets x as the endless total of the powers of a and y as the endless total of the powers of b, with both sizes under one, and asks for the endless total of the powers of ab in terms of x and y. Verified: in the first the exponents themselves form a G.P. with first term 1/2 and ratio 1/2, whose endless total is 1, so the product is 3 to the first power. In the second, x gives a as (x − 1) ÷ x and y gives b as (y − 1) ÷ y, so 1 − ab reduces to (x + y − 1) ÷ (xy) and the answer is xy ÷ (x + y − 1). The second item is the best in the chapter for showing that a sum to infinity is a number like any other and can be solved for.
What the argument turns away (an added construction; §8.6 does not discuss it). Verified on the two boundary ratios: with r = 1 the finite totals are n copies of a and grow without limit, and the formula's denominator is zero; with r = −1 the finite totals alternate between a and 0 forever and settle on nothing, even though they do not grow. The second case is the valuable one, because it shows that failing to grow is not the same as settling.

Figures to have open

A running-total plot: the finite totals drawn as a climbing staircase flattening towards a horizontal line it never touches. This is the topic's single most important image, it carries sections 2, 4 and the second misconception, and the fact that the staircase never meets the line must be visible. Standard schematic; the book prints no such picture.
The same staircase for a negative ratio, closing in on its line from both sides. Standard schematic.
A two-piece bar for the split of section 5, one piece fixed and one piece visibly shrinking. Standard schematic.
A redrawn version of the section's four-column table. The printed table is the only piece of tabular matter in this material and it should be redrawn, not photographed.
No artwork exists in the chapter itself. All sixteen printed pages of pp. 135–150 were opened as images, along with both Supplementary Material pages carrying §8.6: there is no numbered figure anywhere in either, and the only illustration is the portrait on p. 135.

Where this sits in the book

NCERT Class XI Mathematics, Supplementary Material, printed pp. 357–358, §8.6 Infinite G.P. and its Sum, together with Exercise 8.3 on p. 358. This is the numbered section of Chapter 8 that the chapter file itself does not carry.
Chapter 8 §8.4.1, p. 140, which names the infinite geometric series and stops there
Chapter 8 §8.4.2, p. 140, for the finite total this argument is built on
Chapter 8 §8.3, p. 137, where the possibility of an infinite series is first raised