JEE Mains Maths · Sequences and Series

Infinite Geometric Series

The sum to infinity of a GP with ratio between −1 and 1, the series of its squares or alternate terms, and infinite GPs hidden inside exponents, logarithms, double sums and repeated figures.

All Sequences and Series notes JEE Mains Maths strategy

Why this matters

Eighteen PYQs, split evenly. Half give an infinite GP outright, often with the sum of its squares. The other half hide one: in an exponent like cos²x + cos⁴x + …, in a chain of logarithms, or in shapes shrinking step by step. Two ideas cover the page.

Concept 1 of 2: Sum to infinity, and the series of squares

When

|r|<1

, the terms shrink to zero and

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

. The squares of the terms form another infinite GP,

a^2,a^2r^2,\dots

, with sum

\frac{a^2}{1-r^2}

. Dividing that by

S_\infty

cancels one

a

and leaves

\frac{a}{1+r}

. Alternate terms form a GP with ratio

r^2

Definition

$S_\infty=\frac a{1-r}$ , only for $|r|<1$ .
Squares: $\frac{a^2}{1-r^2}$ . Cubes: $\frac{a^3}{1-r^3}$ .
$\frac{\text{sum of squares}}{S_\infty}=\frac a{1+r}$ .
Alternate terms from $ar^k$ : $\frac{ar^k}{1-r^2}$ .

Sum to infinity

S_\infty=\frac{a}{1-r},\quad |r|<1

Worked example

An infinite GP has sum 6, and the sum of the squares of its terms is 12. Find

a

and

r

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2021 · Paper 18 · Q77Moderate

Example 1 · Sequences and Series · Infinite Geometric Series

If the sum of an infinite GP

a,ar,ar^{2},ar^{3},\ldots

is 15 and the sum of the squares of its each term is 150 , then the sum of

ar^{2},ar^{4},ar^{6},\ldots

is:

Check |r| < 1

A quadratic in

r

may give a root with

|r|\ge1

. The sum to infinity does not exist there, so that root is rejected.

Drill 8 more on sum to infinity, and the series of squares

Concept 2 of 2: Infinite GPs inside exponents, logarithms and figures

An exponent

\cos^2x+\cos^4x+\dots

is an infinite GP with ratio

\cos^2x

, summing to

\cot^2x

. A chain

\log x+\log x^{1/3}+\log x^{1/9}+\dots

\log x

times

1+\frac13+\frac19+\dots=\frac32

. A bracket-by-bracket series such as

(a+b)+(a^2+ab+b^2)+\dots

often regroups into two GPs. Figures that shrink by the same factor each step, like triangles joined at midpoints, give a GP of areas or perimeters.

Definition

$\sum_{k\ge1}\cos^{2k}x=\cot^2x$ ; $\sum_{k\ge1}\sin^{2k}x=\tan^2x$ .
$\sum_{k\ge0}\log x^{c^k}=\frac{\log x}{1-c}$ for $|c|<1$ .
Regroup: a sum over brackets can often be written as $\sum a^i$ times $\sum b^j$ , or as two separate GPs.
Midpoint triangles: each area is $\frac14$ of the last, each perimeter $\frac12$ .

The geometric series

1+x+x^2+\dots=\frac{1}{1-x},\quad |x|<1

Worked example

3^{\sin^2x+\sin^4x+\dots}=3

with

0<x<\frac\pi2

. Find

x

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2021 · Paper 1 · Q79Moderate

Example 2 · Sequences and Series · Infinite Geometric Series

e^{\left( \cos^{2}x+\cos^{4}x+\cos^{6}x+ \ldots\infty \right)\log_{e}2}

satisfies the equation

t^{2}- 9t+ 8 = 0

, then the value of

\frac{2\sin x}{\sin x+\sqrt{3}\cos x}\left( 0 <x<\frac{\pi}{2} \right)

is :

The ratio must stay below 1

\cos^2x+\cos^4x+\dots

diverges where

\cos^2x=1

. Check the stated range keeps the ratio strictly between −1 and 1.

Drill 8 more on infinite gps inside exponents, logarithms and figures

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Sum to infinity, and the series of squares
Sum to infinity
$S_\infty=\frac{a}{1-r},\quad |r|<1$
Infinite GPs inside exponents, logarithms and figures
The geometric series
$1+x+x^2+\dots=\frac{1}{1-x},\quad |x|<1$

Watch out for (2)

Check |r| < 1
→ Sum to infinity, and the series of squares
The ratio must stay below 1
→ Infinite GPs inside exponents, logarithms and figures

Test yourself on Sequences and Series

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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