## Series and Their Notations

### Learning Outcomes

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

At a new job, an employee’s starting salary is 32,100. She receives a 2% annual raise. How much will she have earned by the end of 8 years? ## 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 $2+4+6+8+\dots$. This series can also be written in summation notation as $\sum\limits _{k=1}^{\infty} 2k$, 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: $1+0.2+0.04+0.008+0.0016+\dots$ The common ratio is $r=0.2$. As n gets large, the values of 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 $-1<r<1$. ### 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: Determining Whether the Sum of an Infinite Series is Defined Determine whether the sum of each infinite series is defined. 1. $12+8+4+/dots$ 2. $\dfrac{3}{4}+\dfrac{1}{2}+\dfrac{1}{3}+\dots$ 3. $\sum\limits _{k=1}^{\infty}{27}\cdot\left(\dfrac{1}{3}\right)^k$ 4. $\sum\limits _{k=1}^{\infty}{5k}$ ### try it Determine whether the sum of the infinite series is defined. 1. $\dfrac{1}{3}+\dfrac{1}{2}+\dfrac{3}{4}+\dfrac{9}{8}+\cdots$ 2. $24+(-12)+6+(-3)+\dots$ 3. $\sum\limits _{k=1}^{\infty} 15\cdot(-0.3)^k$ ## 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. ${S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}$ We will examine an infinite series with $r=\frac{1}{2}$. What happens to $r^n$ as n increases? \begin{align} &{\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{align} The value of $r^n$ decreases rapidly. What happens for greater values of n? \begin{align} &{\left(\frac{1}{2}\right)}^{10} = \frac{1}{1\text{,}024} \\&{\left(\frac{1}{2}\right)}^{20} = \frac{1}{1\text{,}048\text{,}576} \\&{\left(\frac{1}{2}\right)}^{30} = \frac{1}{1\text{,}073\text{,}741\text{,}824} \end{align} As n gets 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 gives us the 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: $S=\dfrac{{a}_{1}}{1-r}$ ### 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, $S=\dfrac{{a}_{1}}{1-r}$. 4. Simplify to find S. ### Example: Finding the Sum of an Infinite Geometric Series Find the sum, if it exists, for the following: 1. $10+9+8+7+\dots$ 2. $248.6+99.44+39.776+\dots$ 3. $\sum\limits _{k=1}^{\infty}4\text{,}374\cdot\left(-\dfrac{1}{3}\right)^{k-1}$ 4. $\sum\limits _{k=1}^{\infty}\dfrac{1}{9}\cdot\left(\dfrac{4}{3}\right)^{k}$ ### Example: Finding an Equivalent Fraction for a Repeating Decimal Find the equivalent fraction for the repeating decimal $0.\overline{3}$. ### try it Find the sum if it exists. 1. $2+\dfrac{2}{3}+\dfrac{2}{9}+\dots$ 2. $\sum\limits _{k=1}^{\infty}{0.76k+1}$ 3. $\sum\limits _{k=1}^{\infty}\left(-\dfrac{3}{8}\right)^k$ ## Annuities 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 invests50 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 ${a}_{1}=50, r=1.005,$ and $n=72$ into the formula, and simplify to find the value of the annuity after 6 years. ${S}_{72}=\dfrac{50\left(1-{1.005}^{72}\right)}{1 - 1.005}\approx 4\text{,}320.44$ After the last deposit, the couple will have a total of$4,320.44 in the account. Notice, the couple made 72 payments of $50 each for a total of $72\left(50\right) = 3,600$. This means that because of the annuity, the couple earned$720.44 interest in their college fund.

### How To: Given an initial deposit and an interest rate, find the value of an annuity.

1. Determine ${a}_{1}$, the value of the initial deposit.
2. Determine $n$, the number of deposits.
3. Determine $r$.
1. Divide the annual interest rate by the number of times per year that interest is compounded.
2. Add 1 to this amount to find $r$.
4. Substitute values for ${a}_{1},r,$ and $n$
into the formula for the sum of the first $n$ terms of a geometric series, ${S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}$.
5. Simplify to find ${S}_{n}$, the value of the annuity after $n$ deposits.

 sum of the first $n$ terms of an arithmetic series ${S}_{n}=\dfrac{n\left({a}_{1}+{a}_{n}\right)}{2}$ sum of the first $n$ terms of a geometric series ${S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r} , r\ne 1$ 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