We have. A useful series to know about is the harmonic series.
The harmonic series is defined as. This series is interesting because it diverges, but it diverges very slowly. From this table, it is not clear that this series actually diverges. However, we can show analytically that the sequence of partial sums diverges, and therefore the series diverges. To show that the sequence of partial sums diverges, we show that the sequence of partial sums is unbounded.
We begin by writing the first several partial sums:. In the previous section, we stated that convergent sequences are bounded.
Thus, the harmonic series diverges. Since the sum of a convergent infinite series is defined as a limit of a sequence, the algebraic properties for series listed below follow directly from the algebraic properties for sequences. Constant Multiple Rule.
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Because the ratio of each term in this series to the previous term is r , the number r is called the ratio. We refer to a as the initial term because it is the first term in the series.
Therefore, the sequence of partial sums is unbounded and thus diverges. Doing so, we see that. Geometric series sometimes appear in slightly different forms. As long as we can rewrite the series in the form given by Equation, it is a geometric series. Determine whether each of the following geometric series converges or diverges, and if it converges, find its sum.
Writing out the first several terms in the series, we have. If it converges, find its sum. We now turn our attention to a nice application of geometric series. We show how they can be used to write repeating decimals as fractions of integers.
As shown earlier, this sum satisfies. Substituting this expression into the expression above and simplifying, we conclude that. The Koch snowflake is interesting because it has finite area, yet infinite perimeter. Although at first this may seem impossible, recall that you have seen similar examples earlier in the text.
The Surprising Flavor of Infinite Series
Here we use a different technique to show that this series converges. By using partial fractions, we can write. We notice that the middle terms cancel each other out, leaving only the first and last terms.
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In a sense, the series collapses like a spyglass with tubes that disappear into each other to shorten the telescope. For this reason, we call a series that has this property a telescoping series. A telescoping series is a series in which most of the terms cancel in each of the partial sums, leaving only some of the first terms and some of the last terms. We can see this by writing out some of the partial sums.
In particular, we see that. 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.
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.
Follow-up: The Infinite Series and the Mind-Blowing Result.
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.
High marks in maths are the key to your success and future plans. Test yourself and learn more on Siyavula Practice. We derive the formula for calculating the value to which a geometric series converges as follows:. Thereafter the growth of the shrub each year is half of it's growth in the previous year.