6.3: Series

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Learning about mathematical sequences is usually a precursor to learning about mathematical series. A mathematical series is a sequence of numbers that is being added together. The importance of mathematical series cannot be understated. Many equations in the sciences cannot be solved by algebraic methods and must resort to series solutions. The notation for a mathematical series is typically the Greek capital letter sigma: \(\Sigma\). The sigma notation is used as a short-hand method of representing a mathematical series with a particular form.
For example, if we are given the mathematical series:
\(1+5+9+13+17+21\)
This can be represented as follows:
\(\sum_{k=0}^{5} 4 k+1\)
We could also express the same series as:
\(\sum_{k=1}^{6} 4 k-3\)
Both expressions represent the terms being added together. This first example is an example of a finite series because it has a last term. Many mathematical series are infinite series. For example:
\(\sum_{k=1}^{\infty} \frac{1}{2^{k}}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\ldots\)
is an example of an infinite series.
Working with infinite series can be quite useful, but also somewhat confusing. The behavior of an infinite series can be contradictory depending on how you analyze it.

Exercises 6.3
Write out each series in expanded notation.
1) \(\quad \sum_{k=1}^{10} 2 k-5\)
2) \(\quad \sum_{k=3}^{7} 6 k-3\)
3) \(\quad \sum_{k=2}^{9}(-1)^{k}\left(\frac{1}{k}\right)\)
4) \(\quad \sum_{k=0}^{10}(-1)^{k+1}(k-4)\)
5) \(\quad \sum_{k=0}^{4} \frac{k^{2}}{2}\)
6) \(\quad \sum_{k=1}^{8} \frac{k}{3^{k}}\)
Write each series using sigma notation.
7) \(\quad 8+12+16+20+24+28\)
8) \(\quad 5+10+15+20+25+30\)
9) \(\quad 2+9+16+23+\dots+65\)
10) \(\quad 5+8+11+14+\cdots+95\)
11) \(\quad 1+4+9+16+\dots+256\)
12) \(\quad 1+8+27+64+\dots+1331\)
13) \(\quad 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}-\frac{1}{32}\)
14) \(\quad 27-9+3-1+\frac{1}{3}-\frac{1}{9}\)