Infinite Series
Infinite Series
So far we have been working only with finite sums, meaning that whenever we determined the sum of a series, we only considered the sum of the first \(n\) terms. We now consider what happens when we add an infinite number of terms together. Surely if we sum infinitely many numbers, no matter how small they are, the answer goes to infinity? In some cases the answer does indeed go to infinity (like when we sum all the positive integers), but surprisingly there are some cases where the answer is a finite real number.
Optional Investigation: Sum of an infinite series
- Cut a piece of string \(\text{1}\) \(\text{m}\) in length.
- Now cut the piece of string in half and place one half on the desk.
- Cut the other half in half again and put one of the pieces on the desk.
- Repeat this process until the piece of string is too short to cut easily.
- Draw a diagram to illustrate the sequence of lengths of the pieces of string.
- Can this sequence be expressed mathematically? Hint: express the shorter lengths of string as a fraction of the original length of string.
- What is the sum of the lengths of all the pieces of string?
- Predict what would happen if these steps could be repeated infinitely many times.
- Will the sum of the lengths of string ever be greater than \(\text{1}\)?
- What can you conclude?
Example
Question
Complete the table below for the geometric series \(T_{n} = ( \cfrac{1}{2} )^{n}\) and answer the questions that follow:
| Terms | \(S_{n}\) | \(1 - S_{n}\) | |
| \(T_{1}\) | \(\cfrac{1}{2}\) | \(\cfrac{1}{2}\) | \(\cfrac{1}{2}\) |
| \(T_{1} + T_{2}\) | |||
| \(T_{1} + T_{2} + T_{3}\) | |||
| \(T_{1} + T_{2} + T_{3} + T_{4}\) |
- As more and more terms are added, what happens to the value of \(S_{n}\)?
- As more more and more terms are added, what happens to the value of \(1 - S_{n}\)?
- Predict the maximum value of \(S_{n}\) for the sum of infinitely many terms in the series.
Complete the table
| Terms | \(S_{n}\) | \(1 - S_{n}\) | |
| \(T_{1}\) | \(\cfrac{1}{2}\) | \(\cfrac{1}{2}\) | \(\cfrac{1}{2}\) |
| \(T_{1} + T_{2}\) | \(\cfrac{1}{2} + \cfrac{1}{4}\) | \(\cfrac{3}{4}\) | \(\cfrac{1}{4}\) |
| \(T_{1} + T_{2} + T_{3}\) | \(\cfrac{1}{2} + \cfrac{1}{4} + \cfrac{1}{8}\) | \(\cfrac{7}{8}\) | \(\cfrac{1}{8}\) |
| \(T_{1} + T_{2} + T_{3} + T_{4}\) | \(\cfrac{1}{2} + \cfrac{1}{4} + \cfrac{1}{8} + \cfrac{1}{16}\) | \(\cfrac{15}{16}\) | \(\cfrac{1}{16}\) |
Consider the value of \(S_{n}\) and \(1 - S_{n}\)
As more terms in the series are added together, the value of \(S_{n}\) increases:
\[\cfrac{1}{2} \quad < \quad \cfrac{3}{4} \quad < \quad \cfrac{7}{8} \quad < \quad \cdots\]However, by considering \(1 - S_{n}\), we notice that the amount by which \(S_{n}\) increases gets smaller and smaller as more terms are added:
\[\cfrac{1}{2} \quad > \quad \cfrac{1}{4} \quad > \quad \cfrac{1}{8} \quad > \quad \cdots\]We can therefore conclude that the value of \(S_n\) is approaching a maximum value of \(\text{1}\); it is converging to \(\text{1}\).
Write conclusion mathematically
We can conclude that the sum of the series
\[\cfrac{1}{2} + \cfrac{1}{4} + \cfrac{1}{8} + \cdots\]gets closer to 1 (\(S_{n} arrow 1\)) as the number of terms approaches infinity (\(n arrow \infty\)), therefore the series converges.
\[\sum _{i = 1}^{\infty}{ ( \cfrac{1}{2} )^{i}} = 1\]We express the sum of an infinite number of terms of a series as
\[{S}_{\infty }=\sum _{i=1}^{\infty }{T}_{i}\]Convergence and divergence
If the sum of a series gets closer and closer to a certain value as we increase the number of terms in the sum, we say that the series converges. In other words, there is a limit to the sum of a converging series. If a series does not converge, we say that it diverges. The sum of an infinite series usually tends to infinity, but there are some special cases where it does not.
Note the following:
- An arithmetic series never converges: as \(n\) tends to infinity, the series will always tend to positive or negative infinity.
- Some geometric series converge (have a limit) and some diverge (as \(n\) tends to infinity, the series does not tend to any limit or it tends to infinity).
This lesson is part of:
Sequences and Series