Geometric Series

Solved Problems

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} .

Solution.

This is a geometric progression with $q = - \frac{1}{\sqrt{2}} .$ Since the sum of a geometric progression is given by

S_{n} = a_{1} \frac{1 - q^{n}}{1 - q},

we have

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} = \frac{1 - {(- \frac{1}{\sqrt{2}})}^{7}}{1 - (- \frac{1}{\sqrt{2}})} = \frac{1 - \frac{1}{8 \sqrt{2}}}{1 + \frac{1}{\sqrt{2}}} = \frac{\frac{8 \sqrt{2} - 1}{8 \sqrt{2}}}{\frac{\sqrt{2} + 1}{\sqrt{2}}} = \frac{8 \sqrt{2} - 1}{8 (\sqrt{2} + 1)} .

Express the repeating decimal $0, 131313 \dots$ as a rational number.

Solution.

We can write:

0, 131313 \dots = \frac{13}{100} + \frac{13}{10000} + \frac{13}{1000000} + \dots = \frac{13}{100} (1 + \frac{1}{100} + \frac{1}{10000} + \dots) .

Using the formula for the sum of infinite geometric series

S = \sum_{n = 0}^{\infty} q^{n} = \frac{1}{1 - q}

with ratio $q = \frac{1}{100},$ we obtain

0, 131313 \dots = \frac{13}{100} \cdot \frac{1}{1 - \frac{1}{100}} = \frac{13}{100} \cdot \frac{1}{\frac{99}{100}} = \frac{13}{99} .

Show that

1 + \frac{1}{x} + \frac{1}{x^{2}} + \frac{1}{x^{3}} + \frac{1}{x^{4}} + \dots = \frac{x}{x - 1}

assuming $x > 1.$

Solution.

Note that if then In this case, the left side is the sum of an infinite geometric progression. Using the formula

we can write the left side as

so that the formula is proved.

Solve the equation

Solution.

We can write the left side of the equation using the formula for the sum of an infinite geometric series:

Then

The roots of the quadratic equations are

Since the answer is

The second term of an infinite geometric progression () is and the sum of the progression is Determine the first term and ratio of the progression.

Solution.

We use the formula for the sum of an infinite geometric series:

Since the second term of a geometric progression is equal to we have the following system of equations to find the first term and ratio

Solving this system we obtain the following quadratic equation

The equation has two roots:

For each ratio we determine the first terms:

Thus, the problem has two answers: