## 17Calculus - Sigma Notation

### Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

Calculus Tools

### Articles

Before working with definite integrals and infinite series, you need to understand sigma notation. Sigma notation is just a compact way to write sums, i.e. adding a sequence of numbers or terms.
A term using sigma notation is called a series or summation (sum for short).

For example, if we have $$1 + 2 + 3 + 4 + 5$$, we can write this as $$\displaystyle{ \sum_{i=1}^{5}{i} }$$. Let's look at each part of this notation.
1. First, $$\sum$$ just means to add. This is the upper-case Greek letter sigma.
2. The letter i is called the index. It is usually an integer. You will also see other variables used here like n or k.
3. The assignment $$i=1$$ under the sigma gives you the starting value of the index and the index letter.
4. The number above the sigma tells you the ending value of the index. We don't need to repeat 'i=' here. The index is incremented by 1 until it reaches in the ending value. It includes the ending value. For infinite series, you will find $$\infty$$ above the sigma.
5. In this sum, we just have i, so we iterate over the index and add up the results.

In general, we add up more interesting things than just integers. So you will sometimes see the notation $$\displaystyle{ \sum_{i=1}^{n}{a_i} }$$ where $$a_i$$ is some term involving the index i.

Before we go on, let's watch a video. This is a good overview of sigma notation. In this video, he starts by explaining the general notation and then he works several examples.

### PatrickJMT - Summation Notation [10min-15secs]

video by PatrickJMT

Practice

Unless otherwise instructed, evaluate these sums. Give your answers in exact form.

Evaluate $$\displaystyle{ \sum_{n=1}^{5}{ (2n+3) } }$$

Problem Statement

Evaluate $$\displaystyle{ \sum_{n=1}^{5}{ (2n+3) } }$$

Solution

### 2590 video

Log in to rate this practice problem and to see it's current rating.

Evaluate $$\displaystyle{ \sum_{n=1}^{6}{ n^2 } }$$

Problem Statement

Evaluate $$\displaystyle{ \sum_{n=1}^{6}{ n^2 } }$$

Solution

### 2591 video

Log in to rate this practice problem and to see it's current rating.

Evaluate $$\displaystyle{ \sum_{k=0}^{4}{ \frac{1}{k^2+1} } }$$

Problem Statement

Evaluate $$\displaystyle{ \sum_{k=0}^{4}{ \frac{1}{k^2+1} } }$$

158/85

Problem Statement

Evaluate $$\displaystyle{ \sum_{k=0}^{4}{ \frac{1}{k^2+1} } }$$

Solution

A minute or two into this video he realizes that he didn't write down one of the terms and he posts a note about it. Even though the final answer in the video is incorrect, his procedure is good to watch.

### 2592 video

158/85

Log in to rate this practice problem and to see it's current rating.

Evaluate $$\displaystyle{ \sum_{n=1}^{3}{ 2^n } }$$

Problem Statement

Evaluate $$\displaystyle{ \sum_{n=1}^{3}{ 2^n } }$$

Solution

### 2593 video

Log in to rate this practice problem and to see it's current rating.

Evaluate $$\displaystyle{\sum_{k=1}^{6}{\frac{1}{k^2}}}$$

Problem Statement

Evaluate $$\displaystyle{\sum_{k=1}^{6}{\frac{1}{k^2}}}$$

Solution

### 879 video

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

Write$$\displaystyle{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+}$$ $$\displaystyle{\frac{1}{16}+}$$ $$\displaystyle{\frac{1}{32}+}$$ $$\displaystyle{\frac{1}{64}}$$ in sigma notation.

Problem Statement

Write$$\displaystyle{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+}$$ $$\displaystyle{\frac{1}{16}+}$$ $$\displaystyle{\frac{1}{32}+}$$ $$\displaystyle{\frac{1}{64}}$$ in sigma notation.

Solution

### 880 video

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

Evaluate $$\displaystyle{\sum_{r=1}^{8}{(r-1)(r+2)}}$$

Problem Statement

Evaluate $$\displaystyle{\sum_{r=1}^{8}{(r-1)(r+2)}}$$

Solution

### 881 video

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

Evaluate $$\displaystyle{\sum_{i=1}^{4}{(-1)^i i^2}}$$

Problem Statement

Evaluate $$\displaystyle{\sum_{i=1}^{4}{(-1)^i i^2}}$$

Solution

### 882 video

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

You CAN Ace Calculus

### Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - squared identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - double-angle formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }$$

Set 4 - half-angle formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Trig Derivatives

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

Inverse Trig Derivatives

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Trig Integrals

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

### Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

Precalculus

### Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

free ideas to save on bags & supplies

 The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free.

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.