## Series and Their Notations

### Learning Objectives

By the end of this section, you will be able to:

• Use summation notation.
• Use the formula for the sum of the ﬁrst $n$ terms of an arithmetic series.
• Use the formula for the sum of the ﬁrst $n$ terms of a geometric series.
• Use the formula for the sum of an inﬁnite geometric series.
• Solve annuity problems.

### Solution

The problem can be represented by a geometric series with ${a}_{1}=26,750$; $n=5$; and $r=1.016$. Substitute values for ${a}_{1}$, $r$, and $n$ into the formula and simplify to find the total amount earned at the end of 5 years.

$\begin{array}{l}{S}_{n}=\frac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}\hfill \\ {S}_{5}=\frac{26\text{,}750\left(1-{1.016}^{5}\right)}{1 - 1.016}\approx 138\text{,}099.03\hfill \end{array}$

Solution

## Using the Formula for the Sum of an Infinite Geometric Series

Thus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first $n$ terms. An infinite series is the sum of the terms of an infinite sequence. An example of an infinite series is $2+4+6+8+..$.

This series can also be written in summation notation as $\sum _{k=1}^{\infty }2k$, where the upper limit of summation is infinity. Because the terms are not tending to zero, the sum of the series increases without bound as we add more terms. Therefore, the sum of this infinite series is not defined. When the sum is not a real number, we say the series diverges.

## Determining Whether the Sum of an Infinite Geometric Series is Defined

If the terms of an infinite geometric series approach 0, the sum of an infinite geometric series can be defined. The terms in this series approach 0:

$1+0.2+0.04+0.008+0.0016+..$.

The common ratio $r\text{ = 0}\text{.2}$. As $n$ gets very large, the values of ${r}^{n}$ get very small and approach 0. Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to 0, the sum of the terms approaches a finite value. The terms of any infinite geometric series with $-1<r<1$ approach 0; the sum of a geometric series is defined when $-1<r<1$.

### A General Note: Determining Whether the Sum of an Infinite Geometric Series is Defined

The sum of an infinite series is defined if the series is geometric and $-1<r<1$.

### How To: Given the first several terms of an infinite series, determine if the sum of the series exists.

1. Find the ratio of the second term to the first term.
2. Find the ratio of the third term to the second term.
3. Continue this process to ensure the ratio of a term to the preceding term is constant throughout. If so, the series is geometric.
4. If a common ratio, $r$, was found in step 3, check to see if $-1<r<1$ . If so, the sum is defined. If not, the sum is not defined.

### Example 6: Determining Whether the Sum of an Infinite Series is Defined

Determine whether the sum of each infinite series is defined.

1. $\text{12 + 8 + 4 + }\dots$
2. $\frac{3}{4}+\frac{1}{2}+\frac{1}{3}+..$.
3. $\sum _{k=1}^{\infty }27\cdot {\left(\frac{1}{3}\right)}^{k}$
4. $\sum _{k=1}^{\infty }5k$

### Solution

1. The ratio of the second term to the first is $\frac{\text{2}}{\text{3}}$,
which is not the same as the ratio of the third term to the second, $\frac{1}{2}$. The series is not geometric.
2. The ratio of the second term to the first is the same as the ratio of the third term to the second. The series is geometric with a common ratio of $\frac{2}{3}\text{.}$ The sum of the infinite series is defined.
3. The given formula is exponential with a base of $\frac{1}{3}$; the series is geometric with a common ratio of $\frac{1}{3}\text{.}$ The sum of the infinite series is defined.
4. The given formula is not exponential; the series is not geometric because the terms are increasing, and so cannot yield a finite sum.

Determine whether the sum of the infinite series is defined.

### Try It 9

$\frac{1}{3}+\frac{1}{2}+\frac{3}{4}+\frac{9}{8}+..$.

Solution

### Try It 10

$24+\left(-12\right)+6+\left(-3\right)+..$.

Solution

### Try It 11

$\sum _{k=1}^{\infty }15\cdot {\left(-0.3\right)}^{k}$

Solution

## Finding Sums of Infinite Series

When the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite series is related to the formula for the sum of the first $n$ terms of a geometric series.

${S}_{n}=\frac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}$

We will examine an infinite series with $r=\frac{1}{2}$. What happens to ${r}^{n}$ as $n$ increases?

$\begin{array}{l}{\left(\frac{1}{2}\right)}^{2}=\frac{1}{4}\\ {\left(\frac{1}{2}\right)}^{3}=\frac{1}{8}\\ {\left(\frac{1}{2}\right)}^{4}=\frac{1}{16}\end{array}$

The value of ${r}^{n}$ decreases rapidly. What happens for greater values of $n?$

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

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 ${a}_{1}$. This give us a formula for the sum of an infinite geometric series.

### A General Note: Formula for the Sum of an Infinite Geometric Series

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

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

### How To: Given an infinite geometric series, find its sum.

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

### Example 7: Finding the Sum of an Infinite Geometric Series

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

1. $10+9+8+7+\dots$
2. $248.6+99.44+39.776+\text{ }\dots$
3. $\sum _{k=1}^{\infty }4\text{,}374\cdot {\left(-\frac{1}{3}\right)}^{k - 1}$
4. $\sum _{k=1}^{\infty }\frac{1}{9}\cdot {\left(\frac{4}{3}\right)}^{k}$

### Solution

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 $r>1$. The sum does not exist.

### Example 8: Finding an Equivalent Fraction for a Repeating Decimal

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

### Solution

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.

### Try It 12

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

Solution

### Try It 13

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

Solution

### Try It 14

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

Solution

## 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.

We can find the value of the annuity after $n$ deposits using the formula for the sum of the first $n$ terms of a geometric series. In 6 years, there are 72 months, so $n=72$. We can substitute ${a}_{1}=50, r=1.005, \text{and} n=72$ into the formula, and simplify to find the value of the annuity after 6 years.

${S}_{72}=\frac{50\left(1-{1.005}^{72}\right)}{1 - 1.005}\approx 4\text{,}320.44$

### Solution

The value of the initial deposit is $100, so ${a}_{1}=100$. A total of 120 monthly deposits are made in the 10 years, so $n=120$. To find $r$, divide the annual interest rate by 12 to find the monthly interest rate and add 1 to represent the new monthly deposit. $r=1+\frac{0.09}{12}=1.0075$ Substitute ${a}_{1}=100\text{,}r=1.0075\text{,}\text{and}n=120$ into the formula for the sum of the first $n$ terms of a geometric series, and simplify to find the value of the annuity. ${S}_{120}=\frac{100\left(1-{1.0075}^{120}\right)}{1 - 1.0075}\approx 19\text{,}351.43$ So the account has$19,351.43 after the last deposit is made.

At the beginning of each month, $200 is deposited into a retirement fund. The fund earns 6% annual interest, compounded monthly, and paid into the account at the end of the month. How much is in the account if deposits are made for 10 years? Solution ## Key Equations  sum of the first $n$ terms of an arithmetic series ${S}_{n}=\frac{n\left({a}_{1}+{a}_{n}\right)}{2}$ sum of the first $n$ terms of a geometric series ${S}_{n}=\frac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}\cdot r\ne 1$ sum of an infinite geometric series with $-1 ## Key Concepts • The sum of the terms in a sequence is called a series. • A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum. • The sum of the terms in an arithmetic sequence is called an arithmetic series. • The sum of the first [latex]n$ terms of an arithmetic series can be found using a formula. • The sum of the terms in a geometric sequence is called a geometric series. • The sum of the first $n$ terms of a geometric series can be found using a formula. • The sum of an infinite series exists if the series is geometric with $-1<r<1$. • If the sum of an infinite series exists, it can be found using a formula. • An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series. ## Glossary annuity an investment in which the purchaser makes a sequence of periodic, equal payments arithmetic series the sum of the terms in an arithmetic sequence diverge a series is said to diverge if the sum is not a real number geometric series the sum of the terms in a geometric sequence index of summation in summation notation, the variable used in the explicit formula for the terms of a series and written below the sigma with the lower limit of summation infinite series the sum of the terms in an infinite sequence lower limit of summation the number used in the explicit formula to find the first term in a series nth partial sum the sum of the first $n$ terms of a sequence series the sum of the terms in a sequence summation notation a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series upper limit of summation the number used in the explicit formula to find the last term in a series ## Section Exercises 1. What is an $n\text{th}$ partial sum? 2. What is the difference between an arithmetic sequence and an arithmetic series? 3. What is a geometric series? 4. How is finding the sum of an infinite geometric series different from finding the $n\text{th}$ partial sum? 5. What is an annuity? For the following exercises, express each description of a sum using summation notation. 6. The sum of terms ${m}^{2}+3m$ from $m=1$ to $m=5$ 7. The sum from of $n=0$ to $n=4$ of $5n$ 8. The sum of $6k - 5$ from $k=-2$ to $k=1$ 9. The sum that results from adding the number 4 five times For the following exercises, express each arithmetic sum using summation notation. 10. $5+10+15+20+25+30+35+40+45+50$ 11. $10+18+26+\dots +162$ 12. $\frac{1}{2}+1+\frac{3}{2}+2+\dots +4$ For the following exercises, use the formula for the sum of the first $n$ terms of each arithmetic sequence. 13. $\frac{3}{2}+2+\frac{5}{2}+3+\frac{7}{2}$ 14. $19+25+31+\dots +73$ 15. $3.2+3.4+3.6+\dots +5.6$ For the following exercises, express each geometric sum using summation notation. 16. $1+3+9+27+81+243+729+2187$ 17. $8+4+2+\dots +0.125$ 18. $-\frac{1}{6}+\frac{1}{12}-\frac{1}{24}+\dots +\frac{1}{768}$ For the following exercises, use the formula for the sum of the first $n$ terms of each geometric sequence, and then state the indicated sum. 19. $9+3+1+\frac{1}{3}+\frac{1}{9}$ 20. $\sum _{n=1}^{9}5\cdot {2}^{n - 1}$ 21. $\sum _{a=1}^{11}64\cdot {0.2}^{a - 1}$ For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason. 22. $12+18+24+30+..$. 23. $2+1.6+1.28+1.024+..$. 24. $\sum _{m=1}^{\infty }{4}^{m - 1}$ 25. $\underset{\infty }{\overset{k=1}{{\sum }^{\text{ }}}}-{\left(-\frac{1}{2}\right)}^{k - 1}$ For the following exercises, use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of$50. Each month thereafter he increased the previous deposit amount by $20. 26. Graph the arithmetic sequence showing one year of Javier’s deposits. 27. Graph the arithmetic series showing the monthly sums of one year of Javier’s deposits. For the following exercises, use the geometric series ${\sum _{k=1}^{\infty }\left(\frac{1}{2}\right)}^{k}$. 28. Graph the first 7 partial sums of the series. 29. What number does ${S}_{n}$ seem to be approaching in the graph? Find the sum to explain why this makes sense. For the following exercises, find the indicated sum. 30. $\sum _{a=1}^{14}a$ 31. $\sum _{n=1}^{6}n\left(n - 2\right)$ 32. $\sum _{k=1}^{17}{k}^{2}$ 33. $\sum _{k=1}^{7}{2}^{k}$ For the following exercises, use the formula for the sum of the first $n$ terms of an arithmetic series to find the sum. 34. $-1.7+-0.4+0.9+2.2+3.5+4.8$ 35. $6+\frac{15}{2}+9+\frac{21}{2}+12+\frac{27}{2}+15$ 36. $-1+3+7+…+31$ 37. $\sum _{k=1}^{11}\left(\frac{k}{2}-\frac{1}{2}\right)$ For the following exercises, use the formula for the sum of the first $n$ terms of a geometric series to find the partial sum. 38. ${S}_{6}$ for the series $-2 - 10 - 50 - 250..$. 39. ${S}_{7}$ for the series $0.4 - 2+10 - 50..$. 40. $\sum _{k=1}^{9}{2}^{k - 1}$ 41. $\sum _{n=1}^{10}-2\cdot {\left(\frac{1}{2}\right)}^{n - 1}$ For the following exercises, find the sum of the infinite geometric series. 42. $4+2+1+\frac{1}{2}..$. 43. $-1-\frac{1}{4}-\frac{1}{16}-\frac{1}{64}..$. 44. $\underset{\infty }{\overset{k=1}{{\sum }^{\text{ }}}}3\cdot {\left(\frac{1}{4}\right)}^{k - 1}$ 45. $\sum _{n=1}^{\infty }4.6\cdot {0.5}^{n - 1}$ For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate. 46. Deposit amount:$50; total deposits: $60$; interest rate: $5%$, compounded monthly

47. Deposit amount: $150; total deposits: $24$; interest rate: $3%$, compounded monthly 48. Deposit amount:$450; total deposits: $60$; interest rate: $4.5%$, compounded quarterly

49. Deposit amount: $100; total deposits: $120$; interest rate: $10%$, compounded semi-annually 50. The sum of terms $50-{k}^{2}$ from $k=x$ through $7$ is $115$. What is x? 51. Write an explicit formula for ${a}_{k}$ such that $\sum _{k=0}^{6}{a}_{k}=189$. Assume this is an arithmetic series. 52. Find the smallest value of n such that $\sum _{k=1}^{n}\left(3k - 5\right)>100$. 53. How many terms must be added before the series $-1 - 3-5 - 7….\text{ }$ has a sum less than $-75?$ 54. Write $0.\overline{65}$ as an infinite geometric series using summation notation. Then use the formula for finding the sum of an infinite geometric series to convert $0.\overline{65}$ to a fraction. 55. The sum of an infinite geometric series is five times the value of the first term. What is the common ratio of the series? 56. To get the best loan rates available, the Riches want to save enough money to place 20% down on a$160,000 home. They plan to make monthly deposits of $125 in an investment account that offers 8.5% annual interest compounded semi-annually. Will the Riches have enough for a 20% down payment after five years of saving? How much money will they have saved? 57. Karl has two years to save $10,000$ to buy a used car when he graduates. To the nearest dollar, what would his monthly deposits need to be if he invests in an account offering a 4.2% annual interest rate that compounds monthly? 58. Keisha devised a week-long study plan to prepare for finals. On the first day, she plans to study for $1$ hour, and each successive day she will increase her study time by $30$ minutes. How many hours will Keisha have studied after one week? 59. A boulder rolled down a mountain, traveling 6 feet in the first second. Each successive second, its distance increased by 8 feet. How far did the boulder travel after 10 seconds? 60. A scientist places 50 cells in a petri dish. Every hour, the population increases by 1.5%. What will the cell count be after 1 day? 61. A pendulum travels a distance of 3 feet on its first swing. On each successive swing, it travels $\frac{3}{4}$ the distance of the previous swing. What is the total distance traveled by the pendulum when it stops swinging? 62. Rachael deposits$1,500 into a retirement fund each year. The fund earns 8.2% annual interest, compounded monthly. If she opened her account when she was 19 years old, how much will she have by the time she is 55? How much of that amount will be interest earned?