Series

The word ‘series’ in common language implies much the same thing as ‘sequence’. But in mathematics when we talk of a series, we are referring in particular to sums of terms in a sequence, eg. for a sequence of values an, the corresponding series is the sequence of S_n with

S_n=a₁+a₂+⋯+a_(n-1)+a_n
If the terms are in an arithmetic sequence, we call the sum an arithmetic series.

Sigma notation

Most of the series we consider in mathematics are infinite series. This name is used to emphasize the fact that the series contain infinitely many terms. Any sum in the series S_k will be called a partial sum and is given by

S_k=a₁+a₂+⋯+a_(k-1)+a_k
For convenience, this partial sum is written using the sigma notation:

Sigma notation is a concise and convenient way to represent long sums.
Here, the symbol Σ is the Greek capital letter sigma that refers to the initial letter of the word ‘sum’. So, the expression ∑^i=k_i=1 a_i means the sum of all the terms a_i, where i takes the values from 1 to k. We can also write ∑ⁿ_i=m a_i to mean the sum of the terms a_i, where i takes the values from m to n. In such a sum, m is called the lower limit and n the upper limit.

Note that we have ∑⁵_n=1 n=∑⁵_i=1 i. The n and the i just play the role of dummy Variables.
We can also work the other way. Sometimes our sum has a pattern which enables us to Write the sum using sigma notation.

Ken wants to get more exercise so he begins by walking for 20 minutes. Each day for two weeks, he increases the length of time that he walks by 5 minutes. At the end of two weeks, the length of time that he has walked each day is given by the following arithmetic sequence:

20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85
The total length of time that Ken has walked in two weeks is the sum of the terms of this arithmetic sequence:
20+25+30+35+40+45+50+55+60+65+70+75+80+85
This sum is called a series.

Definition
A series is the indicated sum of the terms of a sequence.

The symbol Σ, which is the Greek letter sigma, is used to indicate a sum. The number of minutes that Ken walked on the nth day is a_n=20+(n-1)(5). We can write the sum of the number of minutes that Ken walked in sigma notation:

∑¹⁴_n=1 a_n =∑¹⁴_n=120+(n-1)(5)
The “n=1” below Σ is the value of n for the first term of the series, and the number above Σ is the value of n for the last term of the series. The symbol ∑¹⁴_n=1 a_n can be read as “the sum of a_n for all integral values of n from 1 to 14.”
In expanded form:

For example, we can indicate the sum of the first 50 positive even numbers as ∑⁵⁰_i=1 2i.
∑⁵⁰_i=1 2i=2(1)+2(2)+2(3)+2(4)+⋯+2(50)
=2+4+6+8+⋯+100

Note that in this case, i was used to indicate the number of the term. Although any variable can be used, n, i, and k are the variables most frequently used. Be careful not to confuse the variable i with the imaginary number i.

The series ∑⁵⁰_i=1 2i is an example of a finite series since it is the sum of a finite number of terms. An infinite series is the sum of an infinite number of terms of a sequence. We indicate that a series is infinite by using the symbol for infinity, ∞. For example, we can indicate the sum of all of the positive even numbers as:

∑^∞_i=1 2i=2+4+6+⋯+2i+⋯

Writing a series below in sigma notation. Notice that the instructions specify that the lower limit of summation should be 1 and the variable k. So that would be a good thing to adhere to.
4+11+18+25+ ⋯ for n terms It looks arithmetic, with d=7. To go in the summand, I’ll need a formula for u_n. Or, rather, a formula for u_k, since that’s the index variable.

u_k=u₁+(k -1)d=4+(k -1)⋅7
u_k=7k-3
And now to put that in summation form:

Example 1.
Write the following arithmetic series using sigma notation.

3+10+17+24+31+38

Solution 1:
Write a formula: u_n=u₁+(n-1)d
List the information: u₁=3, d=10-2=7.

n=? (n=6, but leave it as an unknown)
u_n=? (u_n=38, but leave it as an unknown)
Substitute and solve: u_n=3+(n-1)⋅7=3+7n-7
∴ u_n=7n-4
Now let n=i, and write, u_i=7i-4.
Hence, 3+10+17+24+31+38=∑⁶_i=1 (7i-4).