Pre-Calculus Module 8: Series and Sigma Notation PDF

PRE- CALCULUS ============= +-----------------------------------------------------------------------+ | **Development Team of the Module** | | | | **Author:** Cathrina P. Roberonta...

PRE- CALCULUS ============= +-----------------------------------------------------------------------+ | **Development Team of the Module** | | | | **Author:** Cathrina P. Roberonta | | | | Josefina L. Estela | | | | **Editor:** SDO La Union, Learning Resource Quality Assurance Team | | | | **Content Reviewer**: Pamela Gutirrez **Language Reviewer**: Emily O. | | Camat **Illustrator:** Ernesto F. Ramos Jr. | | | | **Design and Layout:** Rogelio C. Runas Jr | | | | **Management Team:** | | | | Atty. Donato D. Balderas Jr. | | | | *Schools Division Superintendent* | | | | Vivian Luz S. Pagatpatan, PhD | | | | *Assistant* *Schools Division Superintendent* | | | | German E. Flora, PhD, *CID Chief* | | | | Virgilio C. Boado, PhD, *EPS in Charge of LRMS* | | | | Erlinda M. Dela Peña, EdD, *EPS in Charge of Mathematics* | | | | Claire P. Toluyen, *Librarian II* | +-----------------------------------------------------------------------+ Department of Education -- SDO La Union ======================================= Office Address: [Flores St. Catbangen, San Fernando City, La Union] ----------------- ----------------------------------------------------------------- Telefax: [072 -- 205 -- 0046] Email Address: [launion\@deped.gov.ph] **Senior High School** This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson. Each SLM is composed of different parts. Each part shall guide you step-bystep as you discover and understand the lesson prepared for you. Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher's assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these. In addition to the material in the main text, notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning. Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task. If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator. Thank you.  We begin by discussing the concept of a sequence. Intuitively, a sequence is an ordered list of objects or events. For instance, the sequence of events at a crime scene is important for understanding the nature of the crime. In this module we will be interested in sequences of a more mathematical nature; mostly we will be interested in sequences of numbers. The sum of the numbers in a sequence can be represented as series. The use of sigma notation will shorten the series and make it more compact in form. It will help us shorten in writing out the long sum of a sequence. The sigma notation will be used to denote a sum. This learning material will provide you with information and activities that will deepen your understanding about Sequence and Series. a. b. c. d. 1. Define sequence and series. 2. Compare sequence and series. 3. Rewrite sigma notation into series and vice versa. 4. Find the sum of a sequence using sigma notation. **Pretest** 1. In the expression ∑^8^~𝑏=3~(2𝑏 + 5), which is the index of summation? A. 8 B. 3 C. 5 D. 2 2. In the expression ∑^8^~𝑞=5~(3𝑞 + 2), which is the upper limit? B. 8 B. 3 C. 5 D. 2 3. Rewrite the series 16 + 25 + 36 + 49 + 64 + 81 using sigma notation. A. ∑^6^~𝑚~=4 𝑚^2^ + 1 B. ∑^6^~𝑚~=1 𝑚^2^ C. ∑^9^~𝑚~=1 𝑚^2^ D. ∑^9^~𝑚~=4 𝑚^2^ 4. Using the series 4 + 5 + 6 + 7 + 8 + 9, express in sigma notation C. ∑^9^~𝑘=4~ 𝑘 B. ∑^6^~𝑘=1~ 𝑘 C. ∑^6^~𝑘=4~ 𝑘^2^ D. ∑^9^~𝑚=1~ 𝑘 5. The series 4 + 16 + 24 + 256 + 1024 can be expressed as \_\_\_\_\_. D. ∑^5^~𝑘~=5 𝑘^2^ B. ∑^5^~𝑘~=2 4^𝑘^ C. ∑^5^~𝑚~=1 4^𝑚^ D. ∑^9^~𝑚~=4 𝑚^2^ 6. Rewrite the expression ∑^5^~𝑎=1~ (5𝑎^2^ − 1) as a sum. E. 4 + 19 + 44 + 79 + 124 B. 4 + 16 + 64 + 256 + 1024 C. 19 + 44 + 79 + 124 + 256 D. 4 + 19 + 44 + 64 + 79 + 124 7. The expression ∑^6^~𝑚=1~ 𝑚(𝑚 + 2) can be written as \_\_\_\_\_. F. 3 + 8 + 15 + 24 + 35 + 48 B. 3 + 8 + 15 + 34 + 45 + 48 C. 1 + 2 + 3 + 4 + 5 + 6 D. 3 + 8 + 11 + 15 + 18 + 24 8. Express ∑^6^~𝑘=1~ (5𝑘^2^ + 4) as a sum. G. 9 +10+11+12+13+14+15 B. 9 + 24 + 49 + 84 + 129 + 184 C. 9 + 24 + 49 + 84 + 175 D. 24 + 49 + 84 + 129 + 184 + 204 9. The expression ∑^5^~𝑘=1~ (2𝑘^2^ + 1) is equivalent to \_\_\_\_\_. H. 3 + 8 + 11 + 14 + 17 B. 3 + 9 + 19 + 33 + 73 C. 9 + 19 + 33 + 51 + 73 D. 3 + 9 + 19 + 33 + 51 10. Rewrite the expression ∑^5^~𝑚=0~(40 − 𝑚^2^) as a series. I. 40 + 39 + 36 + 31 + 15 + 4 B. 39 + 36 + 31 + 24 + 15 + 4 C. 40 + 39 + 36 + 31 + 24 + 15 D. 39 + 4 A. 1, 6, 36, 216, 1296... B. -4 + -12 + -36 + -108 + -324 C. − , − , − , − , − ,  D. − + −  + − + −  + − 12. Which of the following illustrates a series? A. 1, 6, 36, 216, 1296... B. -4 + -6 + -36 + -108 + -324 C. −  , − , −  , − , − , D. −  + − + −  + − + −  13. What is the sum of the arithmetic series 4 + 9 + 14 +... + 64? A. 420 B. 442 C. 452 D. 777 14. What is the sum of the arithmetic series -37 + 63 + 163 + 263? A. 420 B. 442 C. 452 D. 777 15. What is the sum of the geometric series -3 + 18 + 108 + 648? A. 420 B. 442 C. 452 D. 777 1. 1, 6, 36, 216, 1296... \_\_\_\_\_\_\_\_\_\_\_ 2. -1, -6, -36, -216, -1296... \_\_\_\_\_\_\_\_\_\_\_ 3. -4 + -12 + -36 + -108 + -324 \_\_\_\_\_\_\_\_\_\_\_ 4. −  + − + −  + − + −  \_\_\_\_\_\_\_\_\_\_\_ 5. 32, 41, 50, 59, 68... \_\_\_\_\_\_\_\_\_\_\_ 3. , , , ,... \_\_\_\_\_\_\_\_\_\_\_ 4.  , , , , ... \_\_\_\_\_\_\_\_\_\_\_ 5. 0.1, 0.01, 0.001, 0.0001... \_\_\_\_\_\_\_\_\_\_\_ **COMPARE AND CONTRAST** +-----------------------------------+-----------------------------------+ | **What is a sequence?** | **What is a series?** | +===================================+===================================+ | | | +-----------------------------------+-----------------------------------+ | | - A series is a sum of numbers | | | (separated by "+ "or "- " | | | | | | - If a sequence is finite, we | | | will refer to the sum of the | | | terms of the sequence as the | | | series associated with the | | | sequence. | +-----------------------------------+-----------------------------------+ | 1. -11, -13.8, -16. 6, -19.4 | Examples of a series | | | | | 2. -30, -50, -70, -90, -110 | 1. -11 + -13.8 + -16.6 + -19.4 | | | | | 3. 4, 12, 36, 108, 324 | 2. -30 + -50 + -70 + -90 + -110 | | | | | | 3. 4 + 12 + 36 + 108 + 324 | +-----------------------------------+-----------------------------------+ | 1. 3, 11, 19, 27,... | Example: | | | | | 2. , , 2, | 1. -11, -13.8, -16. 6, -19.4 is | | 4, 6, 18,... | a sequence, its associated | | | series is -11 + -13.8 + | | 3. , 1, 2, 4, 8... | -16.6 + -19.4 = - | | | | | - An arithmetic sequence is a | | | sequence in which each term | | | after the first is obtained | 2. -30, -50, -70, -90, -110 is a | | by adding a constant (called | sequence, its associated | | the common difference denoted | series is -30 + -50 + -70 + | | as 𝑑 ) to the preceding term. | -90 + -110 = | | | | | - If the 𝑛th of an arithmetic | | | sequence is 𝑎~𝑛~ and the | | | common difference is 𝑑, then | 3. 4, 12, 36, 108, 324 is a | | | sequence, its associated | | | series is 4 + 12 + 36 + 108 + | | | 324 = | | - A geometric sequence is a | | | sequence in which each term | - The associated arithmetic | | after the first term is | series with 𝑛 terms is given | | obtained by multiplying the | by | | preceding term by a constant | | | (called the | 𝑛 ( 𝑎~1~ + 𝑎~𝑛~) 𝑛 \[2𝑎~1~ + ( 𝑛 | | | − 1 )𝑑\] | | | | | | 𝑆~𝑛~ = = | | - If the 𝑛th of a geometric | | | sequence is 𝑎~𝑛~ and the | 2 2 | | | | | | - The associated geometric | | | series with 𝑛 terms is given | | | by | | | | | | 𝑆~𝑛~ = | | | | | | 𝑛𝑎~𝑎~ 𝑖𝑓 𝑟 = 1 | | | | | | {𝑎~1~( 1− 𝑟 ^𝑛^) | +-----------------------------------+-----------------------------------+ *Examples:* 1. 𝑎~𝑛~ = −  + n  43 5 43 5 37 𝑎2 = − 24 + 3 (2) = 24 𝑎3 = − 4324 + 35 (3) = 7724 43 5 39 𝑎~4~ = − ~24~ + ~3~ (4) = 43 5 𝑎~5~ = − ~24~ + ~3~ (5) = 𝑑 =  − =  𝑆~𝑛~ = 𝑎~1~ + 𝑎~2~ + 𝑎~3~ + 𝑎~4~ + 𝑎~5~ = − +  + +  + =  2. 𝑎~𝑛~ = 21 + 4𝑛 𝑎~1~ = 21 + 4 ( 1 ) = 25 𝑎~2~ = 21 + 4 ( 2) = 29 𝑎~3~ = 21 + 4 ( 13) = 33 𝑎~4~ = 21 + 4 ( 4 ) = 37 𝑎~5~ = 21 + 4 ( 5 ) = 41 𝑑 = 33 − 29 = 4 Associated series: 𝑆~𝑛~ = 𝑎~1~ + 𝑎~2~ + 𝑎~3~ + 𝑎~4~ + 𝑎~5~ = 25 + 29 + 33 + 37 + 41 = 165 𝑎~3~ = −6^3−1^ = −36 𝑎~4~ = −6^4−1^ = −216 𝑎~5~ = −6^5−1^ = −1296 𝑟 = − = 6 𝑆~𝑛~ = 𝑎~1~ + 𝑎~2~ + 𝑎~3~ + 𝑎~4~ + 𝑎~5~ = −1 − 6 − 36 − 216 − 1296 = −1555 𝑎~1~ = −4 ∗ 3^1−1^ = −4 𝑎~2~ = −4 ∗ 3^2−1^ = −12 𝑎~3~ = −4 ∗ 3^3−1^ = −36 𝑎~4~ = −4 ∗ 3^3−1^ = −108 𝑎~5~ = −4 ∗ 3^4−1^ = −324 𝑆~𝑛~ = 𝑎~1~ + 𝑎~2~ + 𝑎~3~ + 𝑎~4~ + 𝑎~5~ = −4 − 12 − 36 − 108 − 324 = −484 **Guided Activity** +-----------------------------------+-----------------------------------+ | 1\. Using the subscripted | | | ***a*** notation, how do you | | | represent the first term, | | | second term and third term | | +-----------------------------------+-----------------------------------+ | 2\. Represent the last term. | | +-----------------------------------+-----------------------------------+ | 3\. Represent the sum of all | | | integers from 1 to 100 using | | | the subscripted ***a*** | | | notation | | +-----------------------------------+-----------------------------------+ | 4\. What is the sum of the | | | wrapped pairs in symbol? | | +-----------------------------------+-----------------------------------+ | 5\. How many pairs are there? | | | What is the relation of these | | | pairs to the number of terms | | | ***n***? | | +-----------------------------------+-----------------------------------+ | 6\. Gauss multiplied this sum | | | (your answer in number 4) times | | | the number of pairs (which is | | | HALF the number of terms in his | | | sequence.) Can you give the | | | product? | | +-----------------------------------+-----------------------------------+ | 7\. What formula did we arrive | | | at? | | +-----------------------------------+-----------------------------------+ | 8\. Rewrite this formula using | | | sigma notation | | +-----------------------------------+-----------------------------------+ So, the sum of 1 + 2 + 3 +... + 98 + 99 + 100 can be written as ∑^100^~𝑖=1~ 𝑖 **Here are more examples:** +-----------------------+-----------------------+-----------------------+ | | +--------+--------+ | 𝑘 | | | | | | | | | | | on | | | | | | +========+========+ | | | | | 1 | 4 | | | | | +--------+--------+ | | | | | 2 | 8 | | | | | +--------+--------+ | | | | | 3 | 12 | | | | | +--------+--------+ | | | | | 4 | 16 | | | | | +--------+--------+ | | +-----------------------+-----------------------+-----------------------+ | | +--------+--------+ | 𝑛=1 | | | | positi | | | | | | | | | | | | | | on | | | | | | +========+========+ | | | | | 1 | -3 | | | | | +--------+--------+ | | | | | 2 | 6 | | | | | +--------+--------+ | | | | | 3 | -9 | | | | | +--------+--------+ | | | | | 4 | 12 | | | | | +--------+--------+ | | +-----------------------+-----------------------+-----------------------+ | | | | +-----------------------+-----------------------+-----------------------+ |  | | | | 𝑛 | | | +-----------------------+-----------------------+-----------------------+ | | | | +-----------------------+-----------------------+-----------------------+  **Activity 3: Give Me Five!** +-----------------+-----------------+-----------------+-----------------+ | 2\. 𝑎~𝑛~ = 33 | *a3 = | *a4 = | *a5 = | | + 5𝑛 | \_\_\_\_\_\_\_\ | \_\_\_\_\_\_\_\ | \_\_\_\_\_\_\_\ | | | _\_\_* | _\_\_* | _\_\_* | +=================+=================+=================+=================+ | 3\. 𝑎~𝑛~ = | *a3 = | *a4 = | *a5 = | | −5.4 − 1.3𝑛 | \_\_\_\_\_\_\_\ | \_\_\_\_\_\_\_\ | \_\_\_\_\_\_\_\ | | | _\_\_* | _\_\_* | _\_\_* | +-----------------+-----------------+-----------------+-----------------+ | 4\. 𝑎~𝑛~ = 8 + | *a3 = | *a4 = | *a5 = | | 10𝑛 | \_\_\_\_\_\_\_\ | \_\_\_\_\_\_\_\ | \_\_\_\_\_\_\_\ | | | _\_\_* | _\_\_* | _\_\_* | +-----------------+-----------------+-----------------+-----------------+ | | *a3 = | *a4 = | *a5 = | | | \_\_\_\_\_\_\_\ | \_\_\_\_\_\_\_\ | \_\_\_\_\_\_\_\ | | | _\_\_* | _\_\_* | _\_\_* | +-----------------+-----------------+-----------------+-----------------+ **Activity 4: Sum It Up!** **DIRECTIONS:** Given the first five terms of the activity 3, identify the common difference or common ratio and give their associative series. Use separate sheet of paper for your answers. 1. 𝑎~𝑛~ = 20 − 7𝑛 2. 𝑎~𝑛~ = −1 + 2𝑛 3. 𝑎~𝑛~ = 2.6 − 2.8𝑛 4. 𝑎~𝑛~ = −34 + 10𝑛 5. 𝑎~𝑛~ = 11 + 7𝑛 1. 0 + 4 + 8 + 12 + 16 Answer: \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 2. 2 + +  + + 1 Answer: \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 3. 1 + 4 + 9 + 16 + 25 + 36 Answer: \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 4. 3 + 9 + 27 + 81 + 243 Answer: \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 5. 0 +  + +  + Answer: \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ **Activity 6: Transform Me!** 1. ∑^9^~𝑛=5~ (30 − 𝑛) Answer:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 2. ∑~𝑚~^5^  ^𝑚^ Answer:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 3. ∑^3^~𝑎=0~ 5^𝑎^ Answer:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 4. ∑^10^~𝑛=5~ 𝑛(𝑛 + 1) Answer:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 5. ∑^4^~𝑛=0~ 3𝑛 Answer:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 1. Rewrite the following so that it starts at x = 0 2. Are these equal? Why or why not? 𝑥=1 𝑥=21 +-------------+-------------+-------------+-------------+-------------+ | **CRITERIA* | **5** | **4** | **3** | **2** | | * | | | | | +=============+=============+=============+=============+=============+ | **Solution* | 90-100% of | Almost all | Most | More than | | * | the steps | (85- 89%) | (75-84%) | 75% of the | | | and | of the | | steps and | | | solutions | steps and | of the | solutions | | | have no | solutions | steps and | have | | | mathematica | have no | solutions | mathematica | | | l | mathematica | have no | l | | | errors. | l | mathematica | errors. | | | | errors. | l | | | | | | errors. | | +-------------+-------------+-------------+-------------+-------------+ | **Mathemati | Correct | Correct | Correct | There is | | cal** | terminology | terminology | terminology | little use, | | | and | and | and | or a lot of | | **Work | notation | notation | notation | inappropria | | and** | are always | are usually | are used, | te | | | used, | used, | making it | use of | | **Notation* | making it | making it | easy to | terminology | | * | easy to | easy to | understand | and | | | understand | understand | what was | notation. | | | what was | what was | done. | | | | done. | done. | | | +-------------+-------------+-------------+-------------+-------------+ | **Neatness | The work is | The work is | The work is | The work | | and | presented | presented | presented | appears | | Organizatio | in a neat, | in a neat, | in an | sloppy and | | n** | clear, | clear, | organized | | | | organized | organized | | unorganized | | | fashion | fashion | fashion but |. | | | that is | that is | may | It | | | easy to | usually | | | | | read. | easy to | be hard to | is | | | | read. | read at | | | | | | times. | hard to | | | | | | know what | | | | | | information | | | | | | goes | | | | | | together. | +-------------+-------------+-------------+-------------+-------------+  1. In the expression ∑^8^~𝑎=2~(5𝑏 + 3), which is the index of summation? A. 8 B. 3 C. 5 D. 2 2. In the expression ∑^5^~𝑚=2~(8𝑚 − 3), which is the upper limit? B. 8 B. 3 C. 5 D. 2 3. Rewrite the series 20 + 25 + 30 + 35 + 40 using sigma notation. C. ∑^9^~𝑚=4~ 4𝑚 B. ∑^5^~𝑚=4~ 5𝑚 C. ∑^8^~𝑚=1~ 𝑚^2^ D. ∑^8^~𝑚=4~ 5𝑚 4. Using the series 0 + 1 + 2 + 3 + 4 + 5, express in sigma notation. D. ∑^5^~𝑘~=1 𝑘 B. ∑^5^~𝑘~=0 𝑘 C. ∑^6^~𝑘~=0 𝑘 D. ∑^6^~𝑚~=1 𝑘 5. The series 15 + 18 + 21 + 24 + 27 + 30 can be expressed as \_\_\_\_\_. E. ∑6𝑘=3 𝑎2 B. ∑6𝑘=5 3𝑘 C. ∑10𝑘=5 3𝑎 D. ∑9𝑚=2 𝑚2 + 2 6. Rewrite the expression ∑^5^~𝑛=1~ (2𝑛^2^ + 2) as a sum. A. 4 + 10 + 20 + 34 + 52 B. 20 + 34 + 52 + 74 + 96 --------------------------- --------------------------- C. 10 + 20 + 34 + 52 + 74 D. 4 + 52 + 74 + 96 + 106 7. The expression ∑^6^~𝑛=1~ 𝑛(𝑛 − 1) can be written as \_\_\_\_\_. +-----------------------------------+-----------------------------------+ | | B. 0 + 2 + 6 + 12 + 20 + 30 | +===================================+===================================+ | 8\. Express ∑^6^~𝑘=1~ (3𝑘^2^ + | D. 0 + 5 + 10 + 15 + 20 + 25 | | 4) as a sum. | | +-----------------------------------+-----------------------------------+ | | B. 7 + 16 + 31 + 52 + 112 + 184 | +-----------------------------------+-----------------------------------+ | C. 7 + 16 + 31 + 52 + 79 + 112 | D. 31 + 52 + 79 + 112 + 151 + 204 | +-----------------------------------+-----------------------------------+ 9. The expression ∑^4^~𝑘=1~ (3𝑘^2^ − 1) is equivalent to \_\_\_\_\_. A. 2 + 11 + 26 + 41 B. 2 + 11 + 26 + 44 C. 2 + 11 + 26 + 43 D. 2 + 11 + 26 + 47 10. Rewrite the expression ∑^5^~𝑎=0~(20 − 𝑎^2^) as a series. B. 19 + 16 + 11 + 4 + (-3) + ( -16) B. 19 + 16 + 11 + 4 + (-5) + ( -16) C. 19 + 16 + 11 + 4 + (-5) + ( -18) D. 19 + 16 + 10 + 4 + (-5) + ( -16) 11. Which of the following illustrates a sequence? I. 1, 6, 36, 216, 1296... II. 32, 41, 50, 59, 68... III\. − , −  , − , − , − ,  IV. − + −  + − + −  + − C. I and II B. II and III C. III and IV D. I and IV 12. Which of the following illustrates a series? I. 1, 6, 36, 216, 1296... II. -4 + -12 + -36 + -108 + -324 III\. − , − , −  , − , − , IV. −  + − + −  + − + −  D. I only B. II only C. II and IV D. III and IV 13. What is the sum of the arithmetic series -10 + -2 + 6+...+46? E. 144 B. 442 C. 244 D. 777 14. What is the sum of the arithmetic series 4 + 9 + 14 +...+64? F. 420 B. 442 C. 452 D. 777 15. What is the sum of the geometric series 81 + 27 + 9 +... + ? B. 442 -- -------- D. 777 *References* ============ **Printed Materials:** Department of Education. (2016). Unit 2: Precalculus, Teacher's Guide (pp. 80-90). Pasig City, Philippines Garces, Ian June L. et al. (2016). Pre-Calculus. Manila, Philippines: Vibal Group, Inc. **Website:** Gauss on Arithmetic Sequences. Retrieved July 24 from Sigma Notation. Retrieved July 23, 2020 from Sigma Notation and Series. Retrieved July 27, 2020 from Summation Notation Worksheet. Retrieved July 23, 2020 from [[http://www.web.pdx.edu/\~sstrand/Math253/HW/SigmaNotationWS. pdf]](http://www.web.pdx.edu/~sstrand/Math253/HW/SigmaNotationWS.pdf) Summation Notation + Work With Sequences. Retrieved July 23, 2020 from [[http://math.wsu.edu/faculty/cjacobs/201/sec1.5suppl.pdf]](http://math.wsu.edu/faculty/cjacobs/201/sec1.5suppl.pdf) The Story of Gauss. Retrieved July 24, 2020 from [[https://www.purplemath.com/modules/series3.html]](https://www.purplemath.com/modules/series3.html) **Software:** **For inquiries or feedback, please write or call:**