Series¶
Suppose $\big\{a_n\big\}_{n=0}^\infty$ is a sequence. It is of course possible to add the first finitely many terms of this sequence,
$$
a_0 + a_1 + a_2 + \cdots + a_k = \sum_{n=0}^k a_n.
$$
But does it make sense to add all infinitely many terms of a sequence and expect a finite answer? This question motivates at least the next 6 sections of the our study.
Definition¶
Given a sequence $\big\{a_n\big\}_{n=0}^\infty$, the corresponding series is given by
$$
\sum_{n=0}^\infty a_n = a_0 + a_1 + a_2 + \cdots .
$$
The $k^\text{th}$ partial sum of the series is
$$
S_k := \sum_{n=0}^k a_n = a_0 + a_1 + a_2 + \cdots + a_k,
$$
for any finite $k \geq 0$.
The partial sums of the series form a sequence
$$
\big\{ S_k \big\}_{k=0}^\infty = \big\{ S_0, S_1, S_2, ..., S_k, ... \big\}.
$$
The sum of the series is given by
$$
\sum_{n=0}^\infty a_n = S = \lim S_k,
$$
provided the limit exists. If the limit does exist, then the series is said to be convergent. Otherwise, the series is called divergent.
1. Example¶
A geometric series is the sum of a geometric sequence,
$$
\sum_{n=0}^\infty a_0 r^n = a_0 + a_0 r + a_0 r^2 + \cdots.
$$
Find a formula for the $k^\text{th}$ partial sum of a geometric series, then determine criteria on $r$ for the series to be convergent.
Video solution
2. Example¶
Show that an arithmetic series with common difference $d \neq 0$ must be divergent.
Solution
An arithmetic series is the sum of the terms of an arithmetic sequence, $a_n = a_0 + d\cdot n$. The partial sums therefore obey
$$
S_{k+1} = S_k + d,\ \ \text{or}\ \ S_{k+1} - S_k = d.
$$
If the arithmetic series were to converge, then $\lim S_{k+1}$ and $\lim S_k$ would both equal to the sum $S$, whence
$$
\lim (S_{k+1} - S_k) = 0.
$$
But $\lim (S_{k+1} - S_k) = d \neq 0$, which is a contradiction. Therefore an arithmetic sequences must be divergent.
3. Example¶
Suppose $\vert x\vert < 1$. Find the sum of the series $\displaystyle y = \sum_{n=0}^\infty x^n$, and plot the graph.
Check your answer
This series is geometric with $a_0 = 1$ and $r = x$. The sum is therefore
$$
\sum_{n=0}^\infty x^n = \frac{1}{1-x},
$$
but only for $\vert x \vert < 1$. The graph is shown below.
4. Example¶
Use a geometric series to write the number $10.1\overline{35}$ as a ratio of integers.
Video solution
5. Example¶
Show that the series is convergent and compute its sum.
$$
\sum_{n=1}^\infty \frac{1}{n(n+1)}
$$
Video solution
6. Example¶
The harmonic series is given by
$$
\sum_{n=1}^\infty \frac{1}{n}.
$$
Show that this series is divergent.
Video solution
Theorem¶
If the series $\sum a_n$ is convergent, then $\lim a_n = 0$.
Proof
Consider the partial sums
$$
\begin{array}{}
S_k & = a_0 + a_1 + \cdots + a_{k-1} + a_k, \\
S_{k-1} & = a_0 + a_1 + \cdots + a_{k-1}.
\end{array}
$$
Taking their difference yields
$$
a_k = S_k - S_{k-1}.
$$
Now, taking the limit,
$$
\lim a_k = \lim \left(S_k - S_{k-1}\right) = \lim S_k - \lim S_{k-1}.
$$
Since the series is assumed to be convergent, then $\lim S_k = \lim S_{k-1} = S$, whence
$$
\lim a_k = S - S = 0.
$$
Caution!
This is a one way thoerem! If $\lim a_n = 0$, it does not guarantee that the corresponding series converges. Can you think of a counter-example?
7. Example¶
Show that the series $\displaystyle \sum_{n=1}^\infty \frac{n^2}{5n^2 + 4}$ diverges.
Check your answer
$$\lim_{n\to\infty} \left( \frac{n^2}{5n^2 + 4} \right) = \frac{1}{5} \neq 0.$$
Therefore, by the Test for Divergence, this series diverges.
Discussion¶
Questions? You can ask them below.