Interval of Convergence for Infinite Series with Ratio Test

In summary, the interval of convergence for the given series is x=2 with a radius of convergence of 4. The ratio test was used to determine this, despite initially appearing inconclusive due to the limit equaling 1. However, the answer key proceeded to solve for x assuming x/4 - 1/2 < 1, which is a valid approach.
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Homework Statement



Find the Interval of Convergence for the given series. Check the endpoint behavior carefully sigma[n=0,inf] (n (x-2)^n)/( (n+1)4^n )

Homework Equations





The Attempt at a Solution



I was following along with the answer key and they used the ratio test...
The only problem is that
[x/4 - 1/2] lim n->inf (n+1)^2 / ((n^2 + 2n) ) = x/4 - 1/2
because the lim n->inf (n+1)^2 / ((n^2 + 2n) )
I thought the ratio test was inconclusive if when you took the limit you got 1?
The answer key than proceeded to solve for x assuming
x/4 - 1/2 < 1
can this be done even though the limit equals one?
 
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My bad... I get it now sorry guys...
 

Related to Interval of Convergence for Infinite Series with Ratio Test

1. What is an infinite series in calculus?

An infinite series is a sum of an infinite number of terms. It is a mathematical concept used in calculus to represent a function as a sum of infinitely many simpler functions. It is often used to solve problems related to limits, integrals, and derivatives.

2. What are the types of convergence for infinite series?

There are three types of convergence for infinite series in calculus: absolute convergence, conditional convergence, and divergence. Absolute convergence means that the series converges to a finite value regardless of the order of the terms. Conditional convergence means that the series only converges if the terms are arranged in a specific order. Divergence means that the series does not converge to a finite value.

3. How do you determine if an infinite series converges or diverges?

There are several tests that can be used to determine if an infinite series converges or diverges. These include the comparison test, ratio test, root test, integral test, and alternating series test. Each test has its own criteria and can be used to determine the convergence or divergence of different types of series.

4. What is the difference between a finite series and an infinite series?

A finite series has a limited number of terms while an infinite series has an unlimited number of terms. A finite series can be calculated and has a defined sum, while an infinite series may or may not have a defined sum and requires special techniques to evaluate.

5. How is calculus used to solve problems related to infinite series?

Calculus is used to find the sum of an infinite series, determine if the series converges or diverges, and to approximate the value of the series if it does converge. It is also used to find the radius and interval of convergence for power series and to analyze the behavior of a function near a point using Taylor and Maclaurin series.

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