Sigma notation - Sequences and Series

Sigma notation

Don't you find it tiring when we express a series with many terms using numerous addition and/or subtraction signs? Don't you wish that we have something to symbolise this action? Well we have a solution, introducing the "Sigma Notation"! In this section, we will learn how to utilise the sigma notation to represent a series, as well as how to evaluate it.

Basic concepts:
Arithmetic sequences
Arithmetic series
Geometric sequences
Geometric series

Related concepts:
Introduction to sequences
Introduction to infinite series

Lessons

Notes:

$\sum_{i=1}^n$ (an equation containing $i$ )

$\sum$ : "Sigma"; summation of $i^{th}$ term to $n^{th}$ term
$i$ : index, a counter for the $i^{th}$ term
$n$ : index of ending term

1.
Evaluate the following arithmetic series:
3.
Evaluate the following geometric series:
6.
Evaluate the following infinite geometric series:
7.
Write the following sum in sigma notation, then evaluate

Sigma notation

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Lessons

Notes:

∑i=1n\sum_{i=1}^n∑i=1n​(an equation containing iii) ∑\sum∑ : "Sigma"; summation of ithi^{th}ith term to nthn^{th}nth term iii : index, a counter for the ithi^{th}ith term nnn : index of ending term

1.

Evaluate the following arithmetic series:

a)

∑i=110i\sum_{i=1}^{10}i∑i=110​i

b)

∑i=15−3(i+1)\sum_{i=1}^{5}-3(i+1)∑i=15​−3(i+1)

2.

Write the following sum in sigma notation, then evaluate 7+9+11+13+...+2057+9+11+13+...+2057+9+11+13+...+205

3.

Evaluate the following geometric series:

a)

∑i=132(12)i\sum_{i=1}^3 2(\frac{1}{2})^i∑i=13​2(21​)i

b)

∑i=120(−3)i+1\sum_{i=1}^{20} (-3)^{i+1}∑i=120​(−3)i+1

4.

Write the following sum in sigma notation, then evaluate −100+10−1+110−1100-100+10-1+\frac{1}{10}-\frac{1}{100}−100+10−1+101​−1001​

5.

Use sigma notation to express S10S_{10}S10​ for −5,10,−20,40,...-5, 10, -20, 40, ...−5,10,−20,40,..., then evaluate

6.

Evaluate the following infinite geometric series:

a)

∑i=1∞3(−5)i−1\sum_{i=1}^{\infty} 3(-5)^{i-1}∑i=1∞​3(−5)i−1

b)

∑i=1∞−72(−23)i\sum_{i=1}^{\infty} -\frac{7}{2}(-\frac{2}{3})^i∑i=1∞​−27​(−32​)i

7.

Write the following sum in sigma notation, then evaluate

a)

4+2+1+...4+2+1+...4+2+1+...

b)

1−2+4−8+...1-2+4-8+...1−2+4−8+...

Sigma notation

Don't just watch, practice makes perfect.

We have over 1850 practice questions in Algebra for you to master.

Algebra
Arithmetic sequences

Algebra
Arithmetic series

Algebra
Geometric sequences

Algebra
Geometric series

or

$\sum_{i=1}^n$ (an equation containing $i$ )

$\sum$ : "Sigma"; summation of $i^{th}$ term to $n^{th}$ term
$i$ : index, a counter for the $i^{th}$ term
$n$ : index of ending term

$\sum_{i=1}^{10}i$

$\sum_{i=1}^{5}-3(i+1)$

Write the following sum in sigma notation, then evaluate
$7+9+11+13+...+205$

$\sum_{i=1}^3 2(\frac{1}{2})^i$

$\sum_{i=1}^{20} (-3)^{i+1}$

Write the following sum in sigma notation, then evaluate
$-100+10-1+\frac{1}{10}-\frac{1}{100}$

Use sigma notation to express $S_{10}$ for $-5, 10, -20, 40, ...$ , then evaluate

$\sum_{i=1}^{\infty} 3(-5)^{i-1}$

$\sum_{i=1}^{\infty} -\frac{7}{2}(-\frac{2}{3})^i$

$4+2+1+...$

$1-2+4-8+...$