Definition 9.9.
\begin{equation*}
\displaystyle\sum_{n=L}^\infty b_n := \lim_{M \to \infty} \displaystyle\sum_{n=L}^M b_n \, .
\end{equation*}
Unit 9.4 Infinite series
No discussion of series would be satisfied if it didn’t answer the question, "Is \(0.9999\ldots\) (repeating) actually equal to 1?" As you can probably guess, it is a matter of definition. However, there is a standard definition, and therefore we can in fact supply an answer (see below).
This definition might require a bit of unpacking. First of all, the colon-equal is right: the symbol \(\displaystyle\sum_{n=L}^\infty b_n\) on the left is not already defined, and we are defining it to be the value on the right. So what we are saying is that the sum of an infinite series is the limit of a certain sequence, called the sequence of partial sums.
Example 9.10.
How does this definition apply to the so-called harmonic series, \(\displaystyle\sum_{n=1}^\infty 1/n\text{?}\) It says that this infinite sum is equal to the limit of the sequence \(\{ H_M \}\text{,}\)where \(H_M\) is the harmonic number \(\displaystyle\sum_{n=1}^M 1/n\text{.}\) The harmonic numbers \(H_M\) are said to be the partial sums of the harmonic series. Interpreting the infinite sum in this way doesn’t tell us whether the limit is defined, or if so, what it is, it just tells us that if we can evaluate the limit \(\displaystyle\lim_{M \to \infty} H_M\text{,}\) this is by definition the sum of the harmonic series. If the limit is undefined, then the sum of the harmonic series is undefined.
Checkpoint 142.
The alternating harmonic series is the series
\begin{equation*}
1 - 1/2 + 1/3 - 1/4 + \cdots
\end{equation*}
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\(\displaystyle\sum_{k=1}^\infty\)
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Write the value of this infinite sum as a limit.
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State your guess as to the general behavior of this limit:
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This sum is defined.
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This sum is undefined because it tends to \(\infty\text{.}\)
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This sum is undefined because it tends to \(-\infty\text{.}\)
PopUp. -
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If you answered that the sum is defined, make a guess as to its value.
Example 9.11.
- Problem
- evaluate \(1 + 1/2 + 1/4 + 1/8 \cdots\text{.}\)
- Solution
- this is the infinite sequence \(\displaystyle\sum_{n=0}^\infty (1/2)^n\text{.}\) The value is the limit of the partial sums \(S_M := \displaystyle\sum_{n=0}^M (1/2)^n\text{.}\) Evaluating these finite sums gives\begin{equation*} S_M = \frac{1 - (1/2)^{M+1}}{1 - 1/2} = 2 - \frac{1}{2^M} \, . \end{equation*}The infinite sum is then \(\displaystyle\lim_{M \to \infty} 2 - (1/2)^M\) which is clearly equal to 2.
