Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

8. Definite Integrals

Riemann Sums

Calculus

8. Definite Integrals

Riemann Sums

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1:51 minutes

Problem 5.1.47b

Textbook Question

Sigma notation Express the following sums using sigma notation. (Answers are not unique.)
(b) 4 + 5 + 6 + 7 + 8 + 9

Verified step by step guidance

Step 1: Understand the problem. The goal is to express the sum 4 + 5 + 6 + 7 + 8 + 9 using sigma notation, which is a compact way to represent summation.

Step 2: Identify the pattern in the sequence. The numbers in the sum are consecutive integers starting from 4 and ending at 9.

Step 3: Define the general term of the sequence. The general term can be written as

k

, where

k

represents each integer in the sequence.

Step 4: Determine the range of the index. The sequence starts at 4 and ends at 9, so the index

k

will range from 4 to 9.

Step 5: Write the sum in sigma notation. The sum can be expressed as

\sum k_{k} = 4 ... 9

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sigma Notation

Sigma notation is a concise way to represent the sum of a sequence of numbers. It uses the Greek letter sigma (Σ) to indicate summation, followed by an expression that defines the terms to be added. The notation typically includes an index of summation, which specifies the starting and ending values for the variable that represents the terms in the sum.

Index of Summation

The index of summation is a variable used in sigma notation to denote the position of each term in the sequence being summed. It usually starts at a specified lower limit and increments by one until it reaches an upper limit. For example, in the sum Σ from i=1 to n, 'i' is the index that takes on integer values from 1 to n, allowing for the systematic addition of terms.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. In the case of the sum 4 + 5 + 6 + 7 + 8 + 9, the common difference is 1. Recognizing that the terms form an arithmetic sequence helps in expressing the sum using sigma notation, as it allows for a general formula to represent the terms based on their position in the sequence.