Table of Contents
What are series used for in real life?
We’ve seen that geometric series can get used to calculate how much money you’ve got in the bank. They can also be used to calculate the amount of medicine in a person’s body, if you know the dosing schedule and amount and how quickly the drug decays in the body.
Why do we study convergence of series?
The main reason you’re learning about infinite series is because later on in your course you will meet power series, which allow you to approximate complicated functions by the simplest functions of all: polynomials. Those convergence tests help you determine where power series make sense.
How sequences and series are applicable in your life?
As we discussed earlier, Sequences and Series play an important role in various aspects of our lives. They help us predict, evaluate and monitor the outcome of a situation or event and help us a lot in decision making.
What is the application of sequence?
There are many applications of sequences. To solve problems involving sequences, it is a good strategy to list the first few terms, and look for a pattern that aids in obtaining the general term. When the general term is found, then one can find any term in the sequence without writing all the preceding terms.
What is convergence and divergence used for?
In the world of economics, finance, and trading, divergence and convergence are terms used to describe the directional relationship of two trends, prices, or indicators.
Why is convergence important in math?
Convergent sequences and series of numbers are often used to obtain various estimates, while in numerical methods they are used for the approximate calculation of the values of functions and constants. In problems of this type, it is important to know the “rate” at which a given sequence converges to its limit.
What are the applications of sequence?
Sequences are useful in our daily lives as well as in higher mathematics. For example, the interest portion of monthly payments made to pay off an automobile or home loan, and the list of maximum daily temperatures in one area for a month are sequences.
What is the condition for a series to be convergent?
The precise statement of the test requires a concept that is used quite often in the study of infinite series. ∣ converges. If a series is convergent but not absolutely convergent, it is called conditionally convergent . = r. If converges absolutely.
What is the divergence test for harmonic series?
Divergence Test. The divergence test does not apply to the harmonic series ∑ n=1∞ 1 n , because lim n→∞ 1 n = 0 . In this case, the divergence test gives no information. It is a common misconception that the “converse” of the divergence test holds, i.e. if the terms go to 0 then the sum converges.
Is the comparison test useful for series analysis?
The comparison test is useful, but intuitively it feels limited. For instance, n n. A refinement of the comparison test, described in the next section, will handle series like this. Instead of comparing to a convergent series using an inequality, it is more flexible to compare to a convergent series using behavior of the terms in the limit.