Activity 8.3.2.
While it is physically impossible to add an infinite collection of numbers, we can, of course, add any finite collection of them. In what follows, we investigate how understanding how to find the \(n\)th partial sum (that is, the sum of the first \(n\) terms) enables us to make sense of the infinite sum.
(a)
Sum the first two numbers in this series. That is, find a numeric value for
\begin{equation*}
\sum_{k=1}^2 \frac{1}{k^2}
\end{equation*}
(b)
Next, add the first three numbers in the series.
(c)
Continue adding terms in this series to complete the list below. Carry each sum to at least 8 decimal places.
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\(\displaystyle \displaystyle\sum_{k=1}^{1} \frac{1}{k^2}=1\)
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\(\displaystyle \displaystyle\sum_{k=1}^{2} \frac{1}{k^2}=\)
-
\(\displaystyle \displaystyle\sum_{k=1}^{3} \frac{1}{k^2}=\)
-
\(\displaystyle \displaystyle\sum_{k=1}^{4} \frac{1}{k^2}=\)
-
\(\displaystyle \displaystyle\sum_{k=1}^{5} \frac{1}{k^2}=\phantom{1.46361111}\)
-
\(\displaystyle \displaystyle\sum_{k=1}^{6} \frac{1}{k^2}=\)
-
\(\displaystyle \displaystyle\sum_{k=1}^{7} \frac{1}{k^2}=\)
-
\(\displaystyle \displaystyle\sum_{k=1}^{8} \frac{1}{k^2}=\)
-
\(\displaystyle \displaystyle\sum_{k=1}^{9} \frac{1}{k^2}=\)
-
\(\displaystyle \displaystyle\sum_{k=1}^{10} \frac{1}{k^2} = \phantom{1.549767731}\)
(d)
The sums in part 8.3.2.c form a sequence whose \(n\)th term is \(S_n = \sum_{k=1}^{n} \frac{1}{k^2}\text{.}\) Based on your calculations in the table, do you think the sequence \(\{S_n\}\) converges or diverges? Explain. How do you think this sequence \(\{S_n\}\) is related to the series \(\sum_{k=1}^{\infty} \frac{1}{k^2}\text{?}\)
