![]() This process is important because it allows us to evaluate, differentiate, and integrate complicated functions by using polynomials. We will use geometric series in the next chapter to write certain functions as polynomials with an infinite number of terms. Example 5.1.1 Finding Terms of Sequences Given by Explicit Formulas Dene sequences a 1,a. The following example shows that it is possible for two different formulas to give sequences with the same terms. We introduce one of the most important types of series: the geometric series. An explicit formula or general formula for a sequence is a rule that shows how the values of a k depend on k. ![]() We also define what it means for a series to converge or diverge. 14.2.6.3: Infinite Series In this section we define an infinite series and show how series are related to sequences.So the sequence is increasing.Also 3n 1+3n < 1foralln which means the sequence is bounded above. We close this section with the Monotone Convergence Theorem, a tool we can use to prove that certain types of sequences converge. The sequence is increasing and bounded above so Theorem 1 applies and says the sequence is convergent. But it is easier to use this Rule: x n n (n+1)/2. We show how to find limits of sequences that converge, often by using the properties of limits for functions discussed earlier. The Triangular Number Sequence is generated from a pattern of dots which form a triangle: By adding another row of dots and counting all the dots we can find the next number of the sequence. 14.2.6.2: Sequences In this section, we introduce sequences and define what it means for a sequence to converge or diverge. ![]() Having defined the necessary tools, we will be able to calculate the area of the Koch snowflake. How do we add an infinite number of terms? Can a sum of an infinite number of terms be finite? To answer these questions, we need to introduce the concept of an infinite series, a sum with infinitely many terms. Consequently, we can express its area as a sum of infinitely many terms. 14.2.6.1: Prelude to Sequence and Series The Koch snowflake is constructed from an infinite number of nonoverlapping equilateral triangles.
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