17Calculus - Sigma Notation
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).
Recommended Books on Amazon | ||
|---|---|---|
  |
  |
  |
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
video by The Organic Chemistry Tutor |
|---|
|
close solution
|
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
video by The Organic Chemistry Tutor |
|---|
|
close solution
|
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 \(\displaystyle{ \sum_{k=0}^{4}{ \frac{1}{k^2+1} } }\)
Final Answer |
|---|
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
video by The Organic Chemistry Tutor |
|---|
Final Answer |
|---|
158/85 |
|
close solution
|
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
video by The Organic Chemistry Tutor |
|---|
|
close solution
|
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 |
|---|
|
close solution
|
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 |
|---|
|
close solution
|
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 |
|---|
|
close solution
|
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 |
|---|
|
close solution
|
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
\(\int{\sin(x)~dx} = -\cos(x)+C\) |
\(\int{\cos(x)~dx} = \sin(x)+C\) | |
\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\) |
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\) | |
\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\) |
\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\) |
To bookmark this page and practice problems, log in to your account or set up a free account.
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.
|
| |
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. |
Shop eBags.com, the leading online retailer of luggage, handbags, backpacks, accessories, and more! |
|---|
Practice Instructions
Unless otherwise instructed, evaluate these sums. Give your answers in exact form.
