13.4 Series and their notations  (Page 5/18)

 Page 5 / 18
$\begin{array}{l}{\left(\frac{1}{2}\right)}^{10}=\frac{1}{1\text{,}024}\hfill \\ {\left(\frac{1}{2}\right)}^{20}=\frac{1}{1\text{,}048\text{,}576}\hfill \\ {\left(\frac{1}{2}\right)}^{30}=\frac{1}{1\text{,}073\text{,}741\text{,}824}\hfill \end{array}$

As $n$ gets very large, ${r}^{n}$ gets very small. We say that, as $n$ increases without bound, ${r}^{n}$ approaches 0. As ${r}^{n}$ approaches 0, $1-{r}^{n}$ approaches 1. When this happens, the numerator approaches $\text{\hspace{0.17em}}{a}_{1}.$ This give us a formula for the sum of an infinite geometric series.

Formula for the sum of an infinite geometric series

The formula for the sum of an infinite geometric series with $-1 is

$S=\frac{{a}_{1}}{1-r}$

Given an infinite geometric series, find its sum.

1. Identify ${a}_{1}$ and $r.$
2. Confirm that $–1
3. Substitute values for ${a}_{1}$ and $r$ into the formula, $S=\frac{{a}_{1}}{1-r}.$
4. Simplify to find $\text{\hspace{0.17em}}S.$

Finding the sum of an infinite geometric series

Find the sum, if it exists, for the following:

1. $10+9+8+7+\dots$
2. $\sum _{k=1}^{\infty }4\text{,}374\cdot {\left(–\frac{1}{3}\right)}^{k–1}$
3. $\sum _{k=1}^{\infty }\frac{1}{9}\cdot {\left(\frac{4}{3}\right)}^{k}$
1. There is not a constant ratio; the series is not geometric.
2. There is a constant ratio; the series is geometric. ${a}_{1}=248.6$ and $r=\frac{99.44}{248.6}=0.4,$ so the sum exists. Substitute ${a}_{1}=248.6$ and $r=0.4$ into the formula and simplify to find the sum:

$\begin{array}{l}S=\frac{{a}_{1}}{1-r}\hfill \\ S=\frac{248.6}{1-0.4}=414.\overline{3}\hfill \end{array}$
3. The formula is exponential, so the series is geometric with $r=–\frac{1}{3}.$ Find ${a}_{1}$ by substituting $k=1$ into the given explicit formula:

${a}_{1}=4\text{,}374\cdot {\left(–\frac{1}{3}\right)}^{1–1}=4\text{,}374$

Substitute ${a}_{1}=4\text{,}374$ and $r=-\frac{1}{3}$ into the formula, and simplify to find the sum:

$\begin{array}{l}S=\frac{{a}_{1}}{1-r}\hfill \\ S=\frac{4\text{,}374}{1-\left(-\frac{1}{3}\right)}=3\text{,}280.5\hfill \end{array}$
4. The formula is exponential, so the series is geometric, but $\text{\hspace{0.17em}}r>1.\text{\hspace{0.17em}}$ The sum does not exist.

Finding an equivalent fraction for a repeating decimal

Find an equivalent fraction for the repeating decimal $0.\overline{3}$

We notice the repeating decimal $0.\overline{3}=0.333...$ so we can rewrite the repeating decimal as a sum of terms.

$0.\overline{3}=0.3+0.03+0.003+...$

Looking for a pattern, we rewrite the sum, noticing that we see the first term multiplied to 0.1 in the second term, and the second term multiplied to 0.1 in the third term.

Notice the pattern; we multiply each consecutive term by a common ratio of 0.1 starting with the first term of 0.3. So, substituting into our formula for an infinite geometric sum, we have

${S}_{n}=\frac{{a}_{1}}{1-r}=\frac{0.3}{1-0.1}=\frac{0.3}{0.9}=\frac{1}{3}.$

Find the sum, if it exists.

$2+\frac{2}{3}+\frac{2}{9}+...$

3

$\sum _{k=1}^{\infty }0.76k+1$

The series is not geometric.

$\sum _{k=1}^{\infty }{\left(-\frac{3}{8}\right)}^{k}$

$-\frac{3}{11}$

Solving annuity problems

At the beginning of the section, we looked at a problem in which a couple invested a set amount of money each month into a college fund for six years. An annuity    is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned. In the example, the couple invests $50 each month. This is the value of the initial deposit. The account paid 6% annual interest , compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account each month by 100.5% to find the value of the account after interest has been added. We can find the value of the annuity right after the last deposit by using a geometric series with ${a}_{1}=50$ and $r=100.5%=1.005.$ After the first deposit, the value of the annuity will be$50. Let us see if we can determine the amount in the college fund and the interest earned.

How look for the general solution of a trig function
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
root under 3-root under 2 by 5 y square
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
cosA\1+sinA=secA-tanA
why two x + seven is equal to nineteen.
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
simplify each radical by removing as many factors as possible (a) √75
how is infinity bidder from undefined?
what is the value of x in 4x-2+3
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
the indicated sum of a sequence is known as