Simplifying an Infinite Series with Partial Fractions

In summary, an infinite series is a mathematical expression that represents the sum of an infinite number of terms, typically written in the form of a1 + a2 + a3 + ... + an + ..., where a1, a2, a3, etc. are the terms and n is the number of terms. Partial fractions, on the other hand, are a technique for breaking down a complex fraction into smaller, simpler fractions. This can be helpful in simplifying infinite series, as it allows them to be broken down into smaller, more manageable parts. The process for simplifying an infinite series with partial fractions involves breaking it down into a ratio of polynomials, finding the partial fractions for each individual polynomial, and then combining them into a
  • #1
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[tex]\Sigma_{n=1}^{ \infty} \frac{1}{(3n-2)(3n+1)} [/tex] I simplified it to partial fractions to (1/3) / (3n-2) - (1/3) / (3n+1) Now what?
 
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  • #2
An infinite number of terms cancel and a finite number of terms don't. Figure out which ones don't. Start writing out terms in the partial fraction expansion for n=1,2,3,4,5... if you need to. You should see the pattern pretty quickly.
 

1. What is an infinite series?

An infinite series is a mathematical expression that represents the sum of an infinite number of terms. It can be written in the form of a1 + a2 + a3 + ... + an + ..., where a1, a2, a3, etc. are the terms of the series and n is the number of terms.

2. What are partial fractions?

Partial fractions are a way to break down a complex fraction into smaller, simpler fractions. This technique is often used to simplify mathematical expressions involving rational functions.

3. How do partial fractions help to simplify an infinite series?

Infinite series can often be written as a ratio of polynomials, which can then be broken down into partial fractions. This simplifies the series by breaking it down into smaller, more manageable parts.

4. What is the process for simplifying an infinite series with partial fractions?

The process involves breaking down the series into a ratio of polynomials, then finding the partial fractions for each individual polynomial. These partial fractions are then combined to form a simpler expression.

5. Are there any limitations to using partial fractions to simplify an infinite series?

Yes, there are limitations. Partial fractions can only be used on series that can be written as a ratio of polynomials. They also cannot be used on series with non-rational functions, such as trigonometric functions.

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