Calculus

Infinite Sequences and Series

Geometric Series

Page 1
Problems 1-2
Page 2
Problems 3-7

A sequence of numbers \(\left\{ {{a_n}} \right\}\) is called a geometric sequence if the quotient of successive terms is a constant, called the common ratio. Thus \({\large\frac{{{a_{n + 1}}}}{{{a_n}}}\normalsize} = q\) or \({a_{n + 1}} = q{a_n}\) for all terms of the sequence. It’s supposed that \(q \ne 0\) and \(q \ne 1.\)

For any geometric sequence:

\[{a_n} = {a_1}{q^{n – 1}}.\]

A geometric series is the indicated sum of the terms of a geometric sequence. For a geometric series with \(q \ne 1,\)

\[
{{S_n} }={ {a_1} + {a_2} + \ldots + {a_n} }={ {a_1}\frac{{1 – {q^n}}}{{1 – q}},\;\;}\kern-0.3pt
{q \ne 1.}
\]

We say that the geometric series converges if the limit \(\lim\limits_{n \to \infty } {S_n}\) exists and is finite.Otherwise the series is said to diverge.

Let \(S = \sum\limits_{n = 0}^\infty {{a_n}} \) \(= {a_1}\sum\limits_{n = 0}^\infty {{q^n}} \) be a geometric series. Then the series converges to \(\large\frac{{{a_1}}}{{1 – q}}\normalsize\) if \(\left| q \right| \lt 1,\) and the series diverges if \(\left| q \right| \gt 1.\)

Solved Problems

Click on problem description to see solution.

 Example 1

Find the sum of the first \(8\) terms of the geometric sequence \(3,6,12, \ldots \)

 Example 2

Find the sum of the series \(1 – 0,37 + 0,{37^2} \) \(- 0,{37^3} + \ldots \)

 Example 3

Find the sum of the series

\[{{S_7} = 1 – \frac{1}{{\sqrt 2 }} + \frac{1}{2} }-{ \frac{1}{{2\sqrt 2 }} + \frac{1}{4} }-{ \frac{1}{{4\sqrt 2 }} + \frac{1}{8}.}\]

 Example 4

Express the repeating decimal \(0,131313 \ldots \) as a rational number.

 Example 5

Show that

\[{1 + \frac{1}{x} }+{ \frac{1}{{{x^2}}} + \frac{1}{{{x^3}}} }+{ \frac{1}{{{x^4}}} + \ldots }={ \frac{x}{{x – 1}} }\]

assuming \(x \gt 1.\)

 Example 6

Solve the equation

\[
{{x^2} – 2{x^3} }+{ 4{x^4} – 8{x^5} + \ldots }={ 2x + 1,\;\;}\kern-0.3pt
{\left| x \right| \lt 1.}
\]

 Example 7

The second term of an infinite geometric progression (\(\left| q \right| \lt 1\)) is \(21\) and the sum of the progression is \(112.\) Determine the first term and ratio of the progression.

Example 1.

Find the sum of the first \(8\) terms of the geometric sequence \(3,6,12, \ldots \)

Solution.

Here \({a_1} = 3\) and \(q = 2.\) For \(n = 8\) we have

\[
{{S_8} = {a_1}\frac{{1 – {q^8}}}{{1 – q}} }
= {3 \cdot \frac{{1 – {2^8}}}{{1 – 2}} }
= {3 \cdot \frac{{1 – 256}}{{\left( { – 1} \right)}} }={ 765.}
\]

Example 2.

Find the sum of the series \(1 – 0,37 + 0,{37^2} \) \(- 0,{37^3} + \ldots \)

Solution.

This is an infinite geometric series with ratio \(q = -0,37.\) Hence, the series converges to

\[
{S = \sum\limits_{n = 0}^\infty {{q^n}} }
= {\frac{1}{{1 – \left( { – 0,37} \right)}} }
= {\frac{1}{{1 + 0,37}} }
= {\frac{1}{{1,37}} }
= {\frac{{100}}{{137}}.}
\]
Page 1
Problems 1-2
Page 2
Problems 3-7