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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