Infinite Series

Series
In this section we simply work on the concept of adding up the numbers belonging to arithmetic and geometric sequences. We call the sum of any sequence of numbers a series.

Some Basics
If we add up the terms of a sequence, we obtain what is called a series. If we only sum a finite amount of terms, we get a finite series. We use the symbol Sn to mean the sum of the first n terms of a sequence {a1;a2;a3;…;an}:

Sn=a1+a2+a3+⋯+an

For example, if we have the following sequence of numbers
1; 4; 9; 25; 36; 49;…

and we wish to find the sum of the first 4 terms, then we write
S4=1+4+9+25=39

The above is an example of a finite series since we are only summing 4 terms.
If we sum infinitely many terms of a sequence, we get an infinite series:
S=a1+a2+a3+⋯

In the case of an infinite series, the number of terms is unknown and simply increases to ∞.

The sum of an infinite series
What could we mean by the sum of the series


Let us try adding up the first few terms and see what happens. If we add up the first two terms we get
½+¼=¾.

The sum of the first three terms is
½+¼+⅛=⅞.

The sum of the first four terms is

And the sum of the first five terms is

These sums of the first terms of the series are called partial sums. The first partial sum is just the first term on its own, so in this case it would be ½. The second partial sum is the sum of the first two terms, giving ¾. The third partial sum is the sum of the first three terms, giving ⅞, and so on.
If we write down the partial sums from this example,

we can see they form the beginning of another infinite sequence. The n-th term of this sequence is the n-th partial sum. We can see that the partial sums here form a sequence that has limit 1. So it would make sense to say that this series has sum 1. We write

In general, we say that an infinite series has a sum if the partial sums form a sequence that has a real limit. If the series is
k=1 ak=a1+a2+a3+a4+⋯

then it has a sum if the sequence of partial sums
(a1, a1+a2, a1+a2+a3, …)

has a limit. If the sequence of partial sums does not have a real limit, we say the series does not have a sum.
Here is another infinite series that has a sum. It is the series

To find the sum of this series, we need to work out the partial sums. For this particular series, the best way to do this is to split each individual term into two parts:

If we do this to each term, the series becomes

and so the n-th partial sum is

As you can see, most of the terms in this expression cancel in pairs. We just get the outermost two terms,

So the sequence of partial sums is

and this sequence has limit 1. We say the infinite series sums to 1, and we write

Here is an example of an infinite series that does not have a sum. The series
k=1 1=1+1+1+1+⋯

has the sequence of partial sums
(1, 2, 3, 4, …)

This sequence of partial sums does not tend to a real limit. It tends to infinity. So the series does not have a sum.

You might have noticed that, whenever we have taken an infinite series with a sum, then the individual terms of the series have tended to zero. This is a general feature of infinite series. But this argument does not work in the opposite direction. It is possible to have a series with individual terms tending to zero, but with no sum.

Figuring out whether a series is convergent or not is harder than it looks. It’s not enough for the terms to be getting smaller as n gets large. For example, the series 1+½+⅓+¼+⅕+…+1/k+⋯ looks like it might converge (i.e. this series has a finite sum), but it does not. A series that doesn’t converge is said to diverge.
The series


is called the harmonic series, and it has terms that tend to zero. But the sequence of partial sums for this series tends to infinity. So this series does not have a sum.

Key Point: The n-th partial sum of a series is the sum of the first n terms. The sequence of partial sums of a series sometimes tends to a real limit. If this happens, we say that this limit is the sum of the series. If not, we say that the series has no sum. A series can have a sum only if the individual terms tend to zero. But there are some series with individual terms tending to zero that do not have sums.

💎 Let’s read post 👉Arithmetic Mean ≥ Geometric Mean ≥ Harmonic Mean (AM ≥ GM ≥ HM)👈.

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