How to tell the difference in series

In summary: You need to know the definition of convergence for a series, and be able to apply it in basic cases. You should be able to compare a new series to a known series to see if it converges. You should know the basic tests like the comparison test, the limit comparison test, the ratio test, the root test, and the integral test. You should be able to use the basic tests to evaluate the convergence of a new series. You should know about absolute and conditional convergence, and be able to apply the alternating series test. You should know about the harmonic series, and the p-series.In summary, for success on a first test in a Calculus 2 series course, it is important to have a solid understanding of the definition
  • #1
Kendall Pecere
4
0
Hey all! I am in Calculus 2, and we are starting to get into series. This may seem like an odd question, but on quizzes I seem to have difficulty identifying the type of series in order to be able to properly work it, and I'd like to have this down before I get to the test. Does anybody have a good way of identify whether a series is arithmetic? Geometric? P-series? Harmonic? Professor hasn't really given any suggestions other than to just kind of eyeball it, and that only works to a certain degree. Thanks in advance for any replies!
 
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  • #2
Moved from HW section as this is not specifically a homework question. @Kendall Pecere, if you post a question in the homework section, you must use the homework template.
 
  • #3
Kendall Pecere said:
Hey all! I am in Calculus 2, and we are starting to get into series. This may seem like an odd question, but on quizzes I seem to have difficulty identifying the type of series in order to be able to properly work it, and I'd like to have this down before I get to the test. Does anybody have a good way of identify whether a series is arithmetic? Geometric? P-series? Harmonic? Professor hasn't really given any suggestions other than to just kind of eyeball it, and that only works to a certain degree. Thanks in advance for any replies!

What is the definition of an arithmetic series? Does your given series (if you have one) satisfy that definition? Same questions for "geometric" or "p-series". I am not sure what definition your instructor uses for "harmonic series", but as far as I know there really is only one such series, up to a multiplicative constant, maybe.
 
  • #4
If you are starting on series, I think it is more likely you get questions regarding convergence-divergence rather than computing the sum of converging series which often require knowledge on power series and Fourier series. Maybe you can be asked to evaluate the sum of geometric and telescopic series, as well as Cauchy products, and double series in easy cases.

I'd say that for a first test, you need to be solid on convergence.
 

Related to How to tell the difference in series

1. How can I tell if a series is convergent or divergent?

To determine if a series is convergent or divergent, you can use several mathematical tests such as the ratio test, the comparison test, or the integral test. These tests involve evaluating the behavior of the terms in the series and comparing them to known convergent or divergent series. If the terms approach zero or behave similarly to a known convergent series, then the series is likely convergent. If the terms do not approach zero or behave similarly to a known divergent series, then the series is likely divergent.

2. What is the difference between an arithmetic series and a geometric series?

An arithmetic series is a series where each term is obtained by adding a constant value to the previous term. For example, the series 1, 4, 7, 10, ... is an arithmetic series with a common difference of 3. A geometric series is a series where each term is obtained by multiplying the previous term by a constant value. For example, the series 1, 3, 9, 27, ... is a geometric series with a common ratio of 3. The main difference between these two types of series is how the terms are related to each other.

3. How do I find the sum of a series?

The sum of a series can be found using the formula S = a/(1-r), where S is the sum, a is the first term, and r is the common ratio (for geometric series) or common difference (for arithmetic series). However, this formula only works for certain types of series, such as geometric or telescoping series. For other types of series, you may need to use techniques such as partial fraction decomposition or integration to find the sum.

4. Can a series have more than one type of convergence?

Yes, a series can exhibit both absolute and conditional convergence. Absolute convergence means that the series converges regardless of the order in which the terms are added, while conditional convergence means that the series only converges if the terms are added in a specific order. For example, the alternating harmonic series (-1)^n/n is conditionally convergent, as it only converges if the positive and negative terms are added in an alternating pattern.

5. How can I use the comparison test to determine the convergence of a series?

The comparison test involves comparing the given series to a known convergent or divergent series. If the given series is smaller than a known convergent series, then it must also converge. If the given series is larger than a known divergent series, then it must also diverge. However, if the given series falls between a known convergent and divergent series, then the comparison test is inconclusive and other tests must be used to determine convergence.

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