The infinite geometric sequence

We add $\displaystyle \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots$ – parts of a square of area 1:

• The partial sums: $\displaystyle {S_1}={0.5}$, $\displaystyle {S_2}={0.75}$, $\displaystyle {S_3}={0.875}$, …
• Each term covers half of what is not covered yet, so the sum gets closer to 1 and never passes it.
• 1 is the sum of the infinite sequence: the partial sums tend to 1.

When $\displaystyle -1<q<1$, the powers $\displaystyle q^n$ tend to 0. Precisely: for every positive number we choose, as small as we like, there is a place from which on all the powers are smaller than it in absolute value.

• $\displaystyle {q}={\frac{1}{2}}$ and the number $\displaystyle 0.001$: $\displaystyle {\left(\frac{1}{2}\right)^{10}}={\frac{1}{1024}<0.001}$, and every power after it is smaller still.
• Every smaller number, say $\displaystyle 0.000001$, has a place further on: $\displaystyle \left(\frac{1}{2}\right)^{20}<0.000001$.
• When $\displaystyle q\geq1$ or $\displaystyle q\leq-1$, $\displaystyle |q^n|$ does not get smaller, so it does not tend to 0.

Write $\displaystyle {S_n}={\frac{a_1(1-q^n)}{1-q}}$. When $\displaystyle -1<q<1$, $\displaystyle q^n$ tends to 0, so $\displaystyle S_n$ tends to $\displaystyle \frac{a_1}{1-q}$:

• The sum is said to converge.
• In the example: $\displaystyle {S}={\frac{\frac{1}{2}}{1-\frac{1}{2}}}={1}$.
• When $\displaystyle q\geq1$ or $\displaystyle q\leq-1$ the sequence has no sum.

A repeating decimal is the sum of an infinite geometric sequence:

• $\displaystyle {0.333\ldots}={\frac{3}{10}+\frac{3}{100}+\frac{3}{1000}+\ldots}={\frac{\frac{3}{10}}{1-\frac{1}{10}}}={\frac{1}{3}}$.
• $\displaystyle {0.1212\ldots}={\frac{12}{100}+\frac{12}{10000}+\ldots}={\frac{\frac{12}{100}}{1-\frac{1}{100}}}={\frac{12}{99}}={\frac{4}{33}}$.
• $\displaystyle {0.1666\ldots}={\frac{1}{10}+\left(\frac{6}{100}+\frac{6}{1000}+\ldots\right)}={\frac{1}{10}+\frac{\frac{6}{100}}{1-\frac{1}{10}}}={\frac{1}{10}+\frac{1}{15}}={\frac{1}{6}}$.

The sequence $\displaystyle 1, (x-1), (x-1)^2, \ldots$ is a geometric sequence with $\displaystyle {q}={x-1}$.

• It has a sum when $\displaystyle -1<x-1<1$, that is $\displaystyle 0<x<2$ (and $\displaystyle x\neq1$, so that $\displaystyle q\neq0$).
• The sum: $\displaystyle {S}={\frac{1}{1-(x-1)}}={\frac{1}{2-x}}$. For example when $\displaystyle {x}={1.5}$: $\displaystyle {S}={2}$.

The big square has area 16. Joining the midpoints of its sides gives a square of half the area – 8; and again, and so on without end.

• The areas $\displaystyle 16, 8, 4, \ldots$ are a geometric sequence with $\displaystyle {q}={\frac{1}{2}}$.
• The sum of all the areas: $\displaystyle {\frac{16}{1-\frac{1}{2}}}={32}$.

Calculate the sum of the infinite geometric sequence $\displaystyle 27, 9, 3, \ldots$

Write $\displaystyle 0.4545\ldots$ as a fraction: work out the sum $\displaystyle \frac{45}{100}+\frac{45}{10000}+\ldots$

For which values of $\displaystyle x$ has the sequence $\displaystyle 1, 2x, 4x^2, \ldots$ a sum? (Apart from $\displaystyle {x}={0}$.)

The sum of an infinite geometric sequence is 12, and its first term is 8.
Find $\displaystyle q$.

The sum of an infinite geometric sequence is 9, and the sum of the squares of its terms is 40.5.
Find $\displaystyle a_1$ and $\displaystyle q$. (A hint: the squares are a geometric sequence too.)