Solving Problems with Harmonic or P-Series - Tips & Tricks

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In summary, the conversation discusses the convergence of a series with the sum of 1/n. It is concluded that the series is both a harmonic series and a p-series with p=1, making it divergent. However, the importance of understanding the conditions for convergence is also mentioned.
  • #1
tony873004
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Homework Statement


From the example:
blah, blah, blah, = 1/n which is divergent (harmonic serics)

The Attempt at a Solution



blah, blah, blah, =1/n which is a p-series with p=1, therefore divergent.

We both get the same answer, but since all the problems in this section are either convergent or divergent, coming up with the right answer doesn't always mean you did it right.
 
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  • #2
Uh what is your question actually?
 
  • #3
You're both right. The sum 1/n is the harmonic series, but is it also a p-series with p=1. Hence, both comments are correct.
 
  • #4
Although It depends what you know about the conditions on p for convergence.
 

FAQ: Solving Problems with Harmonic or P-Series - Tips & Tricks

What is a harmonic or p-series?

A harmonic or p-series is a type of mathematical series where the terms in the series follow a specific pattern, such as the reciprocal of natural numbers. Examples of harmonic series include 1 + 1/2 + 1/3 + 1/4 + ... and 1 + 1/4 + 1/9 + 1/16 + ...

How can harmonic or p-series be used to solve problems?

Harmonic or p-series can be used in problem-solving by applying various techniques and formulas, such as the integral test, comparison test, or ratio test. These methods help determine if a series converges or diverges, which can be useful in solving problems involving infinite sums.

What are some tips for solving problems involving harmonic or p-series?

Here are some tips for solving problems involving harmonic or p-series:

  • Understand the properties and rules of harmonic or p-series.
  • Identify the type of series and choose the appropriate test to determine convergence or divergence.
  • Use algebraic manipulation and known formulas to simplify the series.
  • Check for common patterns and use them to your advantage.
  • Practice and familiarize yourself with different examples and types of problems.

Are there any tricks for solving problems involving harmonic or p-series?

Yes, there are some tricks that can help make solving problems involving harmonic or p-series easier:

  • For harmonic series, if the terms do not have a common ratio, try grouping them into subseries to simplify the problem.
  • For p-series, if the exponent is greater than 1, the series converges. If it is less than or equal to 1, the series diverges.
  • For alternating harmonic series, use the alternating series test to determine convergence or divergence.
  • Remember the comparison test and ratio test can be used to compare a series to a known convergent or divergent series.

How can understanding harmonic or p-series be beneficial in other fields of science?

Understanding harmonic or p-series can be beneficial in fields such as physics, engineering, and economics, where infinite sums are often used to model real-world phenomena. Additionally, the techniques and problem-solving skills acquired from working with harmonic or p-series can be applied to other types of mathematical and scientific problems.

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